\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {b \,e^{2} {\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{B}\right )}{B \left (-a d +b c \right )^{2} g^{3}}-\frac {d e \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (\frac {-A -B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{B}\right )}{B \left (-a d +b c \right )^{2} g^{3}} \]
command
int(1/(b*g*x+a*g)^3/(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| derivativedivides | \(\frac {e \left (\frac {d \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B}-\frac {b e \,{\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (1, 2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 A}{B}\right )}{B}\right )}{\left (a d -c b \right )^{2} g^{3}}\) | \(117\) |
| default | \(\frac {e \left (\frac {d \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B}-\frac {b e \,{\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (1, 2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 A}{B}\right )}{B}\right )}{\left (a d -c b \right )^{2} g^{3}}\) | \(117\) |
| risch | \(\frac {e d \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{g^{3} \left (a d -c b \right )^{2} B}-\frac {e^{2} b \,{\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (1, 2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 A}{B}\right )}{g^{3} \left (a d -c b \right )^{2} B}\) | \(131\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (b g x +a g \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )}\, dx \]________________________________________________________________________________________