\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {45 b^{3} f m \,n^{3}}{8 e x}-\frac {3 b^{3} f^{2} m \,n^{3} \ln \left (x \right )}{8 e^{2}}-\frac {21 b^{2} f m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 e x}+\frac {3 b^{2} f^{2} m \,n^{2} \ln \left (1+\frac {e}{f x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 e^{2}}-\frac {9 b f m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4 e x}+\frac {3 b \,f^{2} m n \ln \left (1+\frac {e}{f x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4 e^{2}}-\frac {f m \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 e x}+\frac {f^{2} m \ln \left (1+\frac {e}{f x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 e^{2}}+\frac {3 b^{3} f^{2} m \,n^{3} \ln \left (f x +e \right )}{8 e^{2}}-\frac {3 b^{3} n^{3} \ln \left (d \left (f x +e \right )^{m}\right )}{8 x^{2}}-\frac {3 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (f x +e \right )^{m}\right )}{4 x^{2}}-\frac {3 b n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{4 x^{2}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (f x +e \right )^{m}\right )}{2 x^{2}}-\frac {3 b^{3} f^{2} m \,n^{3} \polylog \left (2, -\frac {e}{f x}\right )}{4 e^{2}}-\frac {3 b^{2} f^{2} m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e}{f x}\right )}{2 e^{2}}-\frac {3 b \,f^{2} m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (2, -\frac {e}{f x}\right )}{2 e^{2}}-\frac {3 b^{3} f^{2} m \,n^{3} \polylog \left (3, -\frac {e}{f x}\right )}{2 e^{2}}-\frac {3 b^{2} f^{2} m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, -\frac {e}{f x}\right )}{e^{2}}-\frac {3 b^{3} f^{2} m \,n^{3} \polylog \left (4, -\frac {e}{f x}\right )}{e^{2}} \]
command
int((a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)/x^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(46399\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f x +e \right )^{m}\right )}{x^{3}}\, dx \]________________________________________________________________________________________