\[ \int \frac {(h+i x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx \]
Optimal antiderivative \[ \frac {2 i \left (-e i +f h \right ) \expIntegral \left (\frac {a +b \ln \left (c \left (f x +e \right )\right )}{b}\right ) {\mathrm e}^{-\frac {a}{b}}}{b c d \,f^{3}}+\frac {i^{2} \expIntegral \left (\frac {2 a +2 b \ln \left (c \left (f x +e \right )\right )}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}}}{b \,c^{2} d \,f^{3}}+\frac {\left (-e i +f h \right )^{2} \ln \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}{b d \,f^{3}} \]
command
int((i*x+h)^2/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| derivativedivides | \(\frac {-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {c^{2} e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{2} f^{2} h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {2 c e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c f h i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c^{2} e f h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c^{2} f^{3} d}\) | \(200\) |
| default | \(\frac {-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {c^{2} e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{2} f^{2} h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {2 c e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c f h i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c^{2} e f h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c^{2} f^{3} d}\) | \(200\) |
| risch | \(-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{d \,f^{3} c^{2} b}+\frac {e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{3} b}+\frac {h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d f b}+\frac {2 e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{3} c b}-\frac {2 h i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{2} c b}-\frac {2 e h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{2} b}\) | \(219\) |
Maple 2021.1 output
\[ \int \frac {\left (i x +h \right )^{2}}{\left (d f x +d e \right ) \left (b \ln \left (\left (f x +e \right ) c \right )+a \right )}\, dx \]________________________________________________________________________________________