\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {7 b^{2} f m \,n^{2}}{32 e \,x^{2}}-\frac {b^{2} f^{2} m \,n^{2} \ln \left (x \right )}{16 e^{2}}-\frac {3 b f m n \left (a +b \ln \left (c \,x^{n}\right )\right )}{8 e \,x^{2}}+\frac {b \,f^{2} m n \ln \left (1+\frac {e}{f \,x^{2}}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{8 e^{2}}-\frac {f m \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4 e \,x^{2}}+\frac {f^{2} m \ln \left (1+\frac {e}{f \,x^{2}}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4 e^{2}}+\frac {b^{2} f^{2} m \,n^{2} \ln \left (f \,x^{2}+e \right )}{32 e^{2}}-\frac {b^{2} n^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{32 x^{4}}-\frac {b n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{8 x^{4}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{4 x^{4}}-\frac {b^{2} f^{2} m \,n^{2} \polylog \left (2, -\frac {e}{f \,x^{2}}\right )}{16 e^{2}}-\frac {b \,f^{2} m n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e}{f \,x^{2}}\right )}{4 e^{2}}-\frac {b^{2} f^{2} m \,n^{2} \polylog \left (3, -\frac {e}{f \,x^{2}}\right )}{8 e^{2}} \]
command
int((a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^5,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(13825\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{5}}\, dx \]________________________________________________________________________________________