\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {e \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (\frac {-A -B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{B}\right )}{B^{2} \left (-a d +b c \right ) g^{2}}+\frac {-d x -c}{B \left (-a d +b c \right ) g^{2} \left (b x +a \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )} \]
command
int(1/(b*g*x+a*g)^2/(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {d x +c}{\left (a d -c b \right ) B \left (b x +a \right ) g^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}-\frac {e \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{g^{2} B^{2} \left (a d -c b \right )}\) | \(113\) |
| derivativedivides | \(-\frac {e \left (-\frac {1}{\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) B \left (A +B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}+\frac {{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B^{2}}\right )}{\left (a d -c b \right ) g^{2}}\) | \(132\) |
| default | \(-\frac {e \left (-\frac {1}{\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) B \left (A +B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}+\frac {{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B^{2}}\right )}{\left (a d -c b \right ) g^{2}}\) | \(132\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (b g x +a g \right )^{2} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}\, dx \]________________________________________________________________________________________