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