\[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}{2} \]
command
integrate((a*m*x**m+b*n*q*ln(c*x**n)**(-1+q))*(a*x**m+b*ln(c*x**n)**q)/x,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{2} x^{2 m}}{2} + a b x^{m} \log {\left (c x^{n} \right )}^{q} + \frac {b^{2} \log {\left (c x^{n} \right )}^{2 q}}{2} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________