\[ \int \frac {h+i x}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx \]
Optimal antiderivative \[ \frac {i \expIntegral \left (\frac {a +b \ln \left (c \left (f x +e \right )\right )}{b}\right ) {\mathrm e}^{-\frac {a}{b}}}{b c d \,f^{2}}+\frac {\left (-e i +f h \right ) \ln \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}{b d \,f^{2}} \]
command
int((i*x+h)/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| derivativedivides | \(-\frac {\frac {i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {h c f \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c e i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c \,f^{2} d}\) | \(88\) |
| default | \(-\frac {\frac {i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {h c f \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c e i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c \,f^{2} d}\) | \(88\) |
| risch | \(-\frac {i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{2} c b}+\frac {h \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d f b}-\frac {e i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{2} b}\) | \(96\) |
Maple 2021.1 output
\[ \int \frac {i x +h}{\left (d f x +d e \right ) \left (b \ln \left (\left (f x +e \right ) c \right )+a \right )}\, dx \]________________________________________________________________________________________