\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx \]
Optimal antiderivative \[ -\frac {b n}{30 d^{2} \left (e x +d \right )^{5}}-\frac {11 b n}{120 d^{3} \left (e x +d \right )^{4}}-\frac {37 b n}{180 d^{4} \left (e x +d \right )^{3}}-\frac {19 b n}{40 d^{5} \left (e x +d \right )^{2}}-\frac {29 b n}{20 d^{6} \left (e x +d \right )}-\frac {29 b n \ln \left (x \right )}{20 d^{7}}+\frac {a +b \ln \left (c \,x^{n}\right )}{6 d \left (e x +d \right )^{6}}+\frac {a +b \ln \left (c \,x^{n}\right )}{5 d^{2} \left (e x +d \right )^{5}}+\frac {a +b \ln \left (c \,x^{n}\right )}{4 d^{3} \left (e x +d \right )^{4}}+\frac {a +b \ln \left (c \,x^{n}\right )}{3 d^{4} \left (e x +d \right )^{3}}+\frac {a +b \ln \left (c \,x^{n}\right )}{2 d^{5} \left (e x +d \right )^{2}}-\frac {e x \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{7} \left (e x +d \right )}-\frac {\ln \left (1+\frac {d}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{7}}+\frac {49 b n \ln \left (e x +d \right )}{20 d^{7}}+\frac {b n \polylog \left (2, -\frac {d}{e x}\right )}{d^{7}} \]
command
integrate((a+b*ln(c*x**n))/x/(e*x+d)**7,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________