\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x^4} \, dx \]
Optimal antiderivative \[ -\frac {19 b^{2} f m \,n^{2}}{108 e \,x^{2}}+\frac {26 b^{2} f^{2} m \,n^{2}}{27 e^{2} x}+\frac {2 b^{2} f^{3} m \,n^{2} \ln \left (x \right )}{27 e^{3}}-\frac {5 b f m n \left (a +b \ln \left (c \,x^{n}\right )\right )}{18 e \,x^{2}}+\frac {8 b \,f^{2} m n \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 e^{2} x}-\frac {2 b \,f^{3} m n \ln \left (1+\frac {e}{f x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 e^{3}}-\frac {f m \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{6 e \,x^{2}}+\frac {f^{2} m \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 e^{2} x}-\frac {f^{3} m \ln \left (1+\frac {e}{f x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 e^{3}}-\frac {2 b^{2} f^{3} m \,n^{2} \ln \left (f x +e \right )}{27 e^{3}}-\frac {2 b^{2} n^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{27 x^{3}}-\frac {2 b n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (f x +e \right )^{m}\right )}{9 x^{3}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{3 x^{3}}+\frac {2 b^{2} f^{3} m \,n^{2} \polylog \left (2, -\frac {e}{f x}\right )}{9 e^{3}}+\frac {2 b \,f^{3} m n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e}{f x}\right )}{3 e^{3}}+\frac {2 b^{2} f^{3} m \,n^{2} \polylog \left (3, -\frac {e}{f x}\right )}{3 e^{3}} \]
command
int((a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)/x^4,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(13227\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f x +e \right )^{m}\right )}{x^{4}}\, dx \]________________________________________________________________________________________