\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx \]
Optimal antiderivative \[ \frac {d \expIntegral \left (\frac {A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{B}\right ) {\mathrm e}^{-\frac {A}{B}}}{B^{2} \left (-a d +b c \right )^{2} e \,g^{3}}-\frac {2 b \expIntegral \left (\frac {2 A +2 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{B}\right ) {\mathrm e}^{-\frac {2 A}{B}}}{B^{2} \left (-a d +b c \right )^{2} e^{2} g^{3}}+\frac {d x +c}{B \left (-a d +b c \right ) g^{3} \left (b x +a \right )^{2} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )} \]
command
int(1/(b*g*x+a*g)^3/(A+B*ln(e*(d*x+c)/(b*x+a)))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| derivativedivides | \(-\frac {\frac {b \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-2 \,{\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )\right )}{B^{2}}-\frac {e d \left (-\frac {\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )\right )}{B^{2}}}{e^{2} \left (a d -c b \right )^{2} g^{3}}\) | \(268\) |
| default | \(-\frac {\frac {b \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-2 \,{\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )\right )}{B^{2}}-\frac {e d \left (-\frac {\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )\right )}{B^{2}}}{e^{2} \left (a d -c b \right )^{2} g^{3}}\) | \(268\) |
| risch | \(-\frac {d x +c}{\left (a d -c b \right ) B \left (b x +a \right )^{2} g^{3} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )}+\frac {b c d \,{\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \,g^{3} B^{2} \left (a d -c b \right )^{3}}-\frac {2 c \,b^{2} {\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )}{e^{2} g^{3} B^{2} \left (a d -c b \right )^{3}}-\frac {a \,d^{2} {\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \,g^{3} B^{2} \left (a d -c b \right )^{3}}+\frac {2 d b a \,{\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )}{e^{2} g^{3} B^{2} \left (a d -c b \right )^{3}}\) | \(338\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (b g x +a g \right )^{3} \left (B \ln \left (\frac {\left (d x +c \right ) e}{b x +a}\right )+A \right )^{2}}\, dx \]________________________________________________________________________________________