\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx \]
Optimal antiderivative \[ -\frac {b n}{d^{7} x}+\frac {b e n}{30 d^{3} \left (e x +d \right )^{5}}+\frac {17 b e n}{120 d^{4} \left (e x +d \right )^{4}}+\frac {79 b e n}{180 d^{5} \left (e x +d \right )^{3}}+\frac {53 b e n}{40 d^{6} \left (e x +d \right )^{2}}+\frac {103 b e n}{20 d^{7} \left (e x +d \right )}+\frac {103 b e n \ln \left (x \right )}{20 d^{8}}+\frac {-a -b \ln \left (c \,x^{n}\right )}{d^{7} x}-\frac {e \left (a +b \ln \left (c \,x^{n}\right )\right )}{6 d^{2} \left (e x +d \right )^{6}}-\frac {2 e \left (a +b \ln \left (c \,x^{n}\right )\right )}{5 d^{3} \left (e x +d \right )^{5}}-\frac {3 e \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 d^{4} \left (e x +d \right )^{4}}-\frac {4 e \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 d^{5} \left (e x +d \right )^{3}}-\frac {5 e \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 d^{6} \left (e x +d \right )^{2}}+\frac {6 e^{2} x \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{8} \left (e x +d \right )}+\frac {7 e \ln \left (1+\frac {d}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{8}}-\frac {223 b e n \ln \left (e x +d \right )}{20 d^{8}}-\frac {7 b e n \polylog \left (2, -\frac {d}{e x}\right )}{d^{8}} \]
command
integrate((a+b*ln(c*x**n))/x**2/(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 \[ \text {Timed out} \]_____________________________________________________