\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx \]
Optimal antiderivative \[ \frac {28 b d n x}{e^{8}}-\frac {d \left (341 b n +280 a \right ) x}{10 e^{8}}-\frac {7 b n \,x^{2}}{e^{7}}-\frac {28 b d x \ln \left (c \,x^{n}\right )}{e^{8}}-\frac {x^{8} \left (a +b \ln \left (c \,x^{n}\right )\right )}{6 e \left (e x +d \right )^{6}}-\frac {x^{7} \left (8 a +b n +8 b \ln \left (c \,x^{n}\right )\right )}{30 e^{2} \left (e x +d \right )^{5}}-\frac {x^{6} \left (56 a +15 b n +56 b \ln \left (c \,x^{n}\right )\right )}{120 e^{3} \left (e x +d \right )^{4}}-\frac {x^{5} \left (168 a +73 b n +168 b \ln \left (c \,x^{n}\right )\right )}{180 e^{4} \left (e x +d \right )^{3}}+\frac {x^{2} \left (280 a +341 b n +280 b \ln \left (c \,x^{n}\right )\right )}{20 e^{7}}-\frac {x^{4} \left (840 a +533 b n +840 b \ln \left (c \,x^{n}\right )\right )}{360 e^{5} \left (e x +d \right )^{2}}-\frac {x^{3} \left (840 a +743 b n +840 b \ln \left (c \,x^{n}\right )\right )}{90 e^{6} \left (e x +d \right )}+\frac {d^{2} \left (280 a +341 b n +280 b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{10 e^{9}}+\frac {28 b \,d^{2} n \polylog \left (2, -\frac {e x}{d}\right )}{e^{9}} \]
command
integrate(x**8*(a+b*ln(c*x**n))/(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 \[ \text {Timed out} \]_____________________________________________________