\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x) \, dx \]
Optimal antiderivative \[ \frac {5 b n x}{27 e^{2}}+\frac {7 b n \,x^{2}}{108 e}+\frac {b n \,x^{3}}{27}-\frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 e^{2}}-\frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{18 e}-\frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{27}+\frac {2 b n \ln \left (-e x +1\right )}{27 e^{3}}-\frac {2 b n \,x^{3} \ln \left (-e x +1\right )}{27}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (-e x +1\right )}{9 e^{3}}+\frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (-e x +1\right )}{9}-\frac {b n \polylog \left (2, e x \right )}{9 e^{3}}-\frac {b n \,x^{3} \polylog \left (2, e x \right )}{9}+\frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, e x \right )}{3} \]
command
integrate(x**2*(a+b*ln(c*x**n))*polylog(2,e*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {a x^{3} \operatorname {Li}_{1}\left (e x\right )}{9} + \frac {a x^{3} \operatorname {Li}_{2}\left (e x\right )}{3} - \frac {a x^{3}}{27} - \frac {a x^{2}}{18 e} - \frac {a x}{9 e^{2}} + \frac {a \operatorname {Li}_{1}\left (e x\right )}{9 e^{3}} + \frac {2 b n x^{3} \operatorname {Li}_{1}\left (e x\right )}{27} - \frac {b n x^{3} \operatorname {Li}_{2}\left (e x\right )}{9} + \frac {b n x^{3}}{27} - \frac {b x^{3} \log {\left (c x^{n} \right )} \operatorname {Li}_{1}\left (e x\right )}{9} + \frac {b x^{3} \log {\left (c x^{n} \right )} \operatorname {Li}_{2}\left (e x\right )}{3} - \frac {b x^{3} \log {\left (c x^{n} \right )}}{27} + \frac {7 b n x^{2}}{108 e} - \frac {b x^{2} \log {\left (c x^{n} \right )}}{18 e} + \frac {5 b n x}{27 e^{2}} - \frac {b x \log {\left (c x^{n} \right )}}{9 e^{2}} - \frac {2 b n \operatorname {Li}_{1}\left (e x\right )}{27 e^{3}} - \frac {b n \operatorname {Li}_{2}\left (e x\right )}{9 e^{3}} + \frac {b \log {\left (c x^{n} \right )} \operatorname {Li}_{1}\left (e x\right )}{9 e^{3}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________