32.1 Problem number 120

$$\int \frac{x{\log }^2(x)}{(d+ex{)}^4}dx$$

Optimal antiderivative $$-\frac{x}{3d^{2}e(ex+d)}+\frac{x\ln \left(x\right)}{3de{(ex+d)}^2}+\frac{x^2(ex+3d)\ln {\left(x\right)}^2}{6d^2{(ex+d)}^3}-\frac{\ln \left(x\right)\ln (1+\frac{ex}{d})}{3d^2{e}^2}-\frac{\mathrm{polylog}(2,-\frac{ex}{d})}{3d^2{e}^2}$$

command

integrate(x*ln(x)**2/(e*x+d)**4,x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$\frac{(-d-3ex)\log {\left(x\right)}^2}{6d^3{e}^2+18d^2{e}^{3}x+18d{e}^4{x}^2+6e^5{x}^3}+\frac{\left(\{\begin{array}{ll}\frac{x}{d^3}& \text{for}e=0\\ -\frac{1}{2e{(d+ex)}^2}& \text{otherwise}\end{array}\right)\log \left(x\right)}{e}-\frac{\{\begin{array}{ll}\frac{x}{d^3}& \text{for}e=0\\ -\frac{1}{2d^{2}e+2d{e}^{2}x}-\frac{\log \left(x\right)}{2d^{2}e}+\frac{\log (\frac{d}{e}+x)}{2d^{2}e}& \text{otherwise}\end{array}}{e}+\frac{\{\begin{array}{ll}-\frac{1}{e^{3}x}& \text{for}d=0\\ -\frac{1}{2d{e}^2+2e^{3}x}-\frac{\log (d+ex)}{2d{e}^2}& \text{otherwise}\end{array}}{3d}-\frac{\left(\{\begin{array}{ll}\frac{1}{e^{3}x}& \text{for}d=0\\ -\frac{1}{2d{(\frac{d}{x}+e)}^2}& \text{otherwise}\end{array}\right)\log \left(x\right)}{3d}-\frac{2\left(\{\begin{array}{ll}-\frac{1}{e^{2}x}& \text{for}d=0\\ -\frac{\log (d^2+dex)}{de}& \text{otherwise}\end{array}\right)}{3de}+\frac{2\left(\{\begin{array}{ll}\frac{1}{e^{2}x}& \text{for}d=0\\ -\frac{1}{\frac{d^2}{x}+de}& \text{otherwise}\end{array}\right)\log \left(x\right)}{3de}+\frac{\{\begin{array}{ll}-\frac{1}{ex}& \text{for}d=0\\ \frac{\{\begin{array}{ll}\log \left(e\right)\log \left(x\right)+{\mathrm{Li}}_2\left(\frac{d{e}^{i\pi }}{ex}\right)& \text{for}\left|x\right|<1\\ -\log \left(e\right)\log \left(\frac{1}{x}\right)+{\mathrm{Li}}_2\left(\frac{d{e}^{i\pi }}{ex}\right)& \text{for}\frac{1}{\left|x\right|}<1\\ -G_{2,2}^{2,0}\left(\begin{array}{cc}& 1,1\\ 0,0& \end{array}\right|x)\log \left(e\right)+G_{2,2}^{0,2}\left(\begin{array}{cc}1,1& \\ & 0,0\end{array}\right|x)\log \left(e\right)+{\mathrm{Li}}_2\left(\frac{d{e}^{i\pi }}{ex}\right)& \text{otherwise}\end{array}}{d}& \text{otherwise}\end{array}}{3d{e}^2}-\frac{\left(\{\begin{array}{ll}\frac{1}{ex}& \text{for}d=0\\ \frac{\log (\frac{d}{x}+e)}{d}& \text{otherwise}\end{array}\right)\log \left(x\right)}{3d{e}^2}$$