\[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]
Optimal antiderivative \[ \ln \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right ) \]
command
integrate((a*m*x**m+b*n*q*ln(c*x**n)**(-1+q))/x/(a*x**m+b*ln(c*x**n)**q),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \log {\left (\frac {a x^{m}}{b} + \log {\left (c x^{n} \right )}^{q} \right )} & \text {for}\: b \neq 0 \\m \log {\left (x \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________