Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 5 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 5th\ derivative\ of\ function\ {({x}^{2} + 2x + 3)}^{{e}^{(2x)}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x^{2} + 2x + 3)^{{e}^{(2x)}}\\\\ &\color{blue}{The\ 5th\ derivative\ of\ function:} \\=&32{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3) + 480{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3) + \frac{160x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{4}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{160{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{4}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{448x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{544(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + 800{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3) + \frac{3424x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{4336{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{576x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{1632{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} - \frac{320x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{640x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{624(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{1472x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{320x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{384x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{768x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{320{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{384(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{192x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{480x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{4736x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{1952x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{3072x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{1824x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{4128x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{288(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{2496x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{2976{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{2976{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{1344x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{2304{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + 320{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{4}(x^{2} + 2x + 3) + \frac{128x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} - \frac{96x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{192x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{5472x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{11488x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{96(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{1728(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{640x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1920x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{2304x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1920x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{768x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{2304x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{2048x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{5440x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{640x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{2880x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1152x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{640{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{6872{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{128(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{320x^{3}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1152x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{128x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1856x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{6336x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{4032x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{13536x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{6048x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{7296x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{3512{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{320{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{2880x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{768(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960x^{4}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{128(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{2752{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{3840x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{960x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{14976x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{5760x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{960x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{3840x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{6528x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1792x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1760x^{4}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{7040x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{10560x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{7040x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{960x^{4}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{3840x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{5760x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{3840x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{160x^{4}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{640x^{3}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{960x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{640x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{960{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{320{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1760{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{5472{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + 32{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{5}(x^{2} + 2x + 3) + \frac{2336{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{26400x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{19360x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{192x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{2}} - \frac{3648x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{576x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} - \frac{2880x^{4}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{12480x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{192x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} - \frac{20160x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{160{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{14400x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{576x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} + \frac{3520x^{4}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{13824x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{640x^{4}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{2720x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{2688x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{2368x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{4320x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{6560x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{3040x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{192(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} - \frac{3840{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{288(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{2}} + \frac{4248{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{192(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} + \frac{320{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)} + \frac{800{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{576x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} + \frac{15840x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{1120x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{192x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} - \frac{920{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{4432{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} - \frac{576x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} + \frac{5280{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{480x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{2}} - \frac{6888{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{4080x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{9504x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{11064x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} - \frac{6720x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{1920{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{1840{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} - \frac{17064x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{768x^{5}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{3840x^{4}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{4800x^{3}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{960x^{4}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{160x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)} - \frac{1600x^{5}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{8000x^{4}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{7680x^{3}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{8640x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{16000x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{2880x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{640x^{3}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{1120x^{5}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{5600x^{4}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{11200x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{7680x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{16000x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{11200x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{320x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} - \frac{320x^{5}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{1600x^{4}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{3200x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{3200x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{3840x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{8000x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{5600x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{1600x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{32x^{5}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{160x^{4}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{320x^{3}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{320x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{160x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{768{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{1600{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{1120{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{320{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{32{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 5th\ derivative\ of\ function\ {({x}^{2} + 2x + 3)}^{{e}^{(2x)}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x^{2} + 2x + 3)^{{e}^{(2x)}}\\\\ &\color{blue}{The\ 5th\ derivative\ of\ function:} \\=&32{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3) + 480{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3) + \frac{160x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{4}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{160{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{4}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{448x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{544(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + 800{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3) + \frac{3424x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{4336{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{576x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{1632{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} - \frac{320x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{640x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{624(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{1472x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{320x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{384x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{768x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{320{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{384(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{192x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{480x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{4736x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{1952x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{3072x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{1824x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{4128x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{288(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{2496x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{2976{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{2976{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{1344x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{2304{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + 320{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{4}(x^{2} + 2x + 3) + \frac{128x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} - \frac{96x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{192x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{5472x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{11488x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{96(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{1728(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{640x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1920x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{2304x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1920x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{768x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{2304x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{2048x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{5440x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{640x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{2880x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1152x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{640{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{6872{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{128(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)} + \frac{320x^{3}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1152x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{128x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1856x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{6336x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{4032x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{13536x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{6048x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{7296x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{3512{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} + \frac{320{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{3}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{2}} - \frac{2880x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{768(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960x^{4}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{128(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{2752{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{3840x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{960x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{14976x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{5760x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{960x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{3840x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{6528x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1792x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1760x^{4}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{7040x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{10560x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{7040x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{960x^{4}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{3840x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{5760x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{3840x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{160x^{4}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{640x^{3}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{960x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{640x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{960{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + \frac{320{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{2}(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{1760{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{5472{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{384(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(8x)}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} - \frac{960{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} + 32{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln^{5}(x^{2} + 2x + 3) + \frac{2336{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{3}} + \frac{26400x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{19360x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{192x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{2}} - \frac{3648x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{576x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} - \frac{2880x^{4}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{12480x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{192x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} - \frac{20160x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{160{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}ln(x^{2} + 2x + 3)}{(x^{2} + 2x + 3)^{4}} - \frac{14400x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{576x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} + \frac{3520x^{4}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{13824x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{640x^{4}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{2720x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{2688x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{2368x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{4320x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{6560x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{3040x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{192(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} - \frac{3840{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{288(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{2}} + \frac{4248{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{192(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} + \frac{320{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)} + \frac{800{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{576x^{2}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} + \frac{15840x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{1120x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{192x^{3}(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(6x)}}{(x^{2} + 2x + 3)^{3}} - \frac{920{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} + \frac{4432{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} - \frac{576x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{3}} + \frac{5280{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{480x(x^{2} + 2x + 3)^{{e}^{(2x)}}{e}^{(4x)}}{(x^{2} + 2x + 3)^{2}} - \frac{6888{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{4080x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{9504x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{11064x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} - \frac{6720x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{1920{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{1840{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} - \frac{17064x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{768x^{5}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{3840x^{4}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{4800x^{3}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{960x^{4}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} + \frac{160x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)} - \frac{1600x^{5}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{8000x^{4}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{7680x^{3}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{8640x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{4}} - \frac{16000x^{3}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{2880x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{640x^{3}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{3}} + \frac{1120x^{5}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{5600x^{4}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{11200x^{3}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{7680x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{16000x^{2}{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{11200x^{2}{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{320x^{2}{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{2}} - \frac{320x^{5}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{1600x^{4}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{3200x^{3}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{3200x^{2}{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{3840x{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{8000x{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{5600x{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{1600x{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{32x^{5}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{160x^{4}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{320x^{3}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{320x^{2}{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{160x{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{768{e}^{(2x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{1600{e}^{(4x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{1120{e}^{(6x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} - \frac{320{e}^{(8x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}} + \frac{32{e}^{(10x)}(x^{2} + 2x + 3)^{{e}^{(2x)}}}{(x^{2} + 2x + 3)^{5}}\\ \end{split}\end{equation} \]