\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx \]
Optimal antiderivative \[ \frac {21 a \,b^{2} e m \,n^{2} x}{4 f}-\frac {45 b^{3} e m \,n^{3} x}{8 f}+\frac {3 b^{3} m \,n^{3} x^{2}}{4}+\frac {21 b^{3} e m \,n^{2} x \ln \left (c \,x^{n}\right )}{4 f}-\frac {9 b^{2} m \,n^{2} x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{8}-\frac {9 b e m n x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4 f}+\frac {3 b m n \,x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4}+\frac {e m x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 f}-\frac {m \,x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{4}+\frac {3 b^{3} e^{2} m \,n^{3} \ln \left (f x +e \right )}{8 f^{2}}-\frac {3 b^{3} n^{3} x^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{8}+\frac {3 b^{2} n^{2} x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (f x +e \right )^{m}\right )}{4}-\frac {3 b n \,x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{4}+\frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (f x +e \right )^{m}\right )}{2}-\frac {3 b^{2} e^{2} m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {f x}{e}\right )}{4 f^{2}}+\frac {3 b \,e^{2} m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {f x}{e}\right )}{4 f^{2}}-\frac {e^{2} m \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (1+\frac {f x}{e}\right )}{2 f^{2}}-\frac {3 b^{3} e^{2} m \,n^{3} \polylog \left (2, -\frac {f x}{e}\right )}{4 f^{2}}+\frac {3 b^{2} e^{2} m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {f x}{e}\right )}{2 f^{2}}-\frac {3 b \,e^{2} m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (2, -\frac {f x}{e}\right )}{2 f^{2}}-\frac {3 b^{3} e^{2} m \,n^{3} \polylog \left (3, -\frac {f x}{e}\right )}{2 f^{2}}+\frac {3 b^{2} e^{2} m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, -\frac {f x}{e}\right )}{f^{2}}-\frac {3 b^{3} e^{2} m \,n^{3} \polylog \left (4, -\frac {f x}{e}\right )}{f^{2}} \]
command
int(x*(a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(44991\) |
Maple 2021.1 output
\[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x \ln \left (d \left (f x +e \right )^{m}\right )\, dx \]________________________________________________________________________________________