\[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx \]
Optimal antiderivative \[ -\frac {7 b n x}{e^{7}}+\frac {\left (223 b n +140 a \right ) x}{20 e^{7}}+\frac {7 b x \ln \left (c \,x^{n}\right )}{e^{7}}-\frac {x^{7} \left (a +b \ln \left (c \,x^{n}\right )\right )}{6 e \left (e x +d \right )^{6}}-\frac {x^{6} \left (7 a +b n +7 b \ln \left (c \,x^{n}\right )\right )}{30 e^{2} \left (e x +d \right )^{5}}-\frac {x^{5} \left (42 a +13 b n +42 b \ln \left (c \,x^{n}\right )\right )}{120 e^{3} \left (e x +d \right )^{4}}-\frac {x^{2} \left (140 a +153 b n +140 b \ln \left (c \,x^{n}\right )\right )}{40 e^{6} \left (e x +d \right )}-\frac {x^{4} \left (210 a +107 b n +210 b \ln \left (c \,x^{n}\right )\right )}{360 e^{4} \left (e x +d \right )^{3}}-\frac {x^{3} \left (420 a +319 b n +420 b \ln \left (c \,x^{n}\right )\right )}{360 e^{5} \left (e x +d \right )^{2}}-\frac {d \left (140 a +223 b n +140 b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{20 e^{8}}-\frac {7 b d n \polylog \left (2, -\frac {e x}{d}\right )}{e^{8}} \]
command
integrate(x**7*(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 \[ \text {Timed out} \]_____________________________________________________