\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e \log \left (f x^m\right )\right )} \, dx \]
Optimal antiderivative \[ -\frac {b n}{e m x}-\frac {b \,{\mathrm e}^{\frac {d}{e m}} n \left (f \,x^{m}\right )^{\frac {1}{m}} \expIntegral \left (\frac {-d -e \ln \left (f \,x^{m}\right )}{e m}\right ) \left (d +e \ln \left (f \,x^{m}\right )\right )}{e^{2} m^{2} x}+\frac {{\mathrm e}^{\frac {d}{e m}} \left (f \,x^{m}\right )^{\frac {1}{m}} \expIntegral \left (\frac {-d -e \ln \left (f \,x^{m}\right )}{e m}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{e m x} \]
command
int((a+b*ln(c*x^n))/x^2/(d+e*ln(f*x^m)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2296\) |
Maple 2021.1 output
\[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \ln \left (f \,x^{m}\right )+d \right ) x^{2}}\, dx \]________________________________________________________________________________________