\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx \]
Optimal antiderivative \[ 2 a b m n x -4 b^{2} m \,n^{2} x +2 b m n \left (-b n +a \right ) x +4 b^{2} m n x \ln \left (c \,x^{n}\right )-m x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}-\frac {2 b e m n \left (-b n +a \right ) \ln \left (f x +e \right )}{f}-2 a b n x \ln \left (d \left (f x +e \right )^{m}\right )+2 b^{2} n^{2} x \ln \left (d \left (f x +e \right )^{m}\right )-2 b^{2} n x \ln \left (c \,x^{n}\right ) \ln \left (d \left (f x +e \right )^{m}\right )+x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )-\frac {2 b^{2} e m n \ln \left (c \,x^{n}\right ) \ln \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {f x}{e}\right )}{f}-\frac {2 b^{2} e m \,n^{2} \polylog \left (2, -\frac {f x}{e}\right )}{f}+\frac {2 b e m n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {f x}{e}\right )}{f}-\frac {2 b^{2} e m \,n^{2} \polylog \left (3, -\frac {f x}{e}\right )}{f} \]
command
int((a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(10356\) |
Maple 2021.1 output
\[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )\, dx \]________________________________________________________________________________________