\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{4} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{4 b n}-\frac {m \left (a +b \ln \left (c \,x^{n}\right )\right )^{4} \ln \left (1+\frac {f \,x^{2}}{e}\right )}{4 b n}-\frac {m \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \polylog \left (2, -\frac {f \,x^{2}}{e}\right )}{2}+\frac {3 b m n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (3, -\frac {f \,x^{2}}{e}\right )}{4}-\frac {3 b^{2} m \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (4, -\frac {f \,x^{2}}{e}\right )}{4}+\frac {3 b^{3} m \,n^{3} \polylog \left (5, -\frac {f \,x^{2}}{e}\right )}{8} \]
command
int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(77072\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x}\, dx \]________________________________________________________________________________________