\[ \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (-d g +e f \right )^{2} \left (e x +d \right ) \expIntegral \left (\frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{b n}\right ) {\mathrm e}^{-\frac {a}{b n}} \left (c \left (e x +d \right )^{n}\right )^{-\frac {1}{n}}}{b \,e^{3} n}+\frac {2 g \left (-d g +e f \right ) \left (e x +d \right )^{2} \expIntegral \left (\frac {2 a +2 b \ln \left (c \left (e x +d \right )^{n}\right )}{b n}\right ) {\mathrm e}^{-\frac {2 a}{b n}} \left (c \left (e x +d \right )^{n}\right )^{-\frac {2}{n}}}{b \,e^{3} n}+\frac {g^{2} \left (e x +d \right )^{3} \expIntegral \left (\frac {3 a +3 b \ln \left (c \left (e x +d \right )^{n}\right )}{b n}\right ) {\mathrm e}^{-\frac {3 a}{b n}} \left (c \left (e x +d \right )^{n}\right )^{-\frac {3}{n}}}{b \,e^{3} n} \]
command
int((g*x+f)^2/(a+b*ln(c*(e*x+d)^n)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1889\) |
Maple 2021.1 output
\[ \int \frac {\left (g x +f \right )^{2}}{b \ln \left (c \left (e x +d \right )^{n}\right )+a}\, dx \]________________________________________________________________________________________