\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx \]
Optimal antiderivative \[ \frac {b n \,x^{2}}{2 e m}-\frac {b n \,x^{2} \expIntegral \left (\frac {2 d +2 e \ln \left (f \,x^{m}\right )}{e m}\right ) \left (d +e \ln \left (f \,x^{m}\right )\right ) {\mathrm e}^{-\frac {2 d}{e m}} \left (f \,x^{m}\right )^{-\frac {2}{m}}}{e^{2} m^{2}}+\frac {x^{2} \expIntegral \left (\frac {2 d +2 e \ln \left (f \,x^{m}\right )}{e m}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right ) {\mathrm e}^{-\frac {2 d}{e m}} \left (f \,x^{m}\right )^{-\frac {2}{m}}}{e m} \]
command
int(x*(a+b*ln(c*x^n))/(d+e*ln(f*x^m)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2350\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x}{e \ln \left (f \,x^{m}\right )+d}\, dx \]________________________________________________________________________________________