\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx \]
Optimal antiderivative \[ -12 a \,b^{2} m \,n^{2} x +18 b^{3} m \,n^{3} x -6 b^{2} m \,n^{2} \left (-b n +a \right ) x -18 b^{3} m \,n^{2} x \ln \left (c \,x^{n}\right )+6 b m n x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}-m x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}+\frac {6 b^{2} e m \,n^{2} \left (-b n +a \right ) \ln \left (f x +e \right )}{f}+6 a \,b^{2} n^{2} x \ln \left (d \left (f x +e \right )^{m}\right )-6 b^{3} n^{3} x \ln \left (d \left (f x +e \right )^{m}\right )+6 b^{3} n^{2} x \ln \left (c \,x^{n}\right ) \ln \left (d \left (f x +e \right )^{m}\right )-3 b n x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )+x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (f x +e \right )^{m}\right )+\frac {6 b^{3} e m \,n^{2} \ln \left (c \,x^{n}\right ) \ln \left (1+\frac {f x}{e}\right )}{f}-\frac {3 b e m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (1+\frac {f x}{e}\right )}{f}+\frac {6 b^{3} e m \,n^{3} \polylog \left (2, -\frac {f x}{e}\right )}{f}-\frac {6 b^{2} e m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {f x}{e}\right )}{f}+\frac {3 b e m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (2, -\frac {f x}{e}\right )}{f}+\frac {6 b^{3} e m \,n^{3} \polylog \left (3, -\frac {f x}{e}\right )}{f}-\frac {6 b^{2} e m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, -\frac {f x}{e}\right )}{f}+\frac {6 b^{3} e m \,n^{3} \polylog \left (4, -\frac {f x}{e}\right )}{f} \]
command
int((a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(39644\) |
Maple 2021.1 output
\[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f x +e \right )^{m}\right )\, dx \]________________________________________________________________________________________