\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx \]
Optimal antiderivative \[ -\frac {b n}{4 d^{7} x^{2}}+\frac {7 b e n}{d^{8} x}-\frac {b \,e^{2} n}{30 d^{4} \left (e x +d \right )^{5}}-\frac {23 b \,e^{2} n}{120 d^{5} \left (e x +d \right )^{4}}-\frac {34 b \,e^{2} n}{45 d^{6} \left (e x +d \right )^{3}}-\frac {14 b \,e^{2} n}{5 d^{7} \left (e x +d \right )^{2}}-\frac {131 b \,e^{2} n}{10 d^{8} \left (e x +d \right )}-\frac {131 b \,e^{2} n \ln \left (x \right )}{10 d^{9}}+\frac {-a -b \ln \left (c \,x^{n}\right )}{2 d^{7} x^{2}}+\frac {7 e \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{8} x}+\frac {e^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{6 d^{3} \left (e x +d \right )^{6}}+\frac {3 e^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{5 d^{4} \left (e x +d \right )^{5}}+\frac {3 e^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 d^{5} \left (e x +d \right )^{4}}+\frac {10 e^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 d^{6} \left (e x +d \right )^{3}}+\frac {15 e^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 d^{7} \left (e x +d \right )^{2}}-\frac {21 e^{3} x \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{9} \left (e x +d \right )}-\frac {28 e^{2} \ln \left (1+\frac {d}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{9}}+\frac {341 b \,e^{2} n \ln \left (e x +d \right )}{10 d^{9}}+\frac {28 b \,e^{2} n \polylog \left (2, -\frac {d}{e x}\right )}{d^{9}} \]
command
integrate((a+b*ln(c*x**n))/x**3/(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 \[ \text {Timed out} \]_____________________________________________________