\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {2 b^{2} n^{2}}{d^{2} x}-\frac {2 b n \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{2} x}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{d^{2} x}+\frac {e^{2} x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{d^{3} \left (e x +d \right )}+\frac {2 e \ln \left (1+\frac {d}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{d^{3}}-\frac {2 b e n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{d^{3}}-\frac {4 b e n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {d}{e x}\right )}{d^{3}}-\frac {2 b^{2} e \,n^{2} \polylog \left (2, -\frac {e x}{d}\right )}{d^{3}}-\frac {4 b^{2} e \,n^{2} \polylog \left (3, -\frac {d}{e x}\right )}{d^{3}} \]
command
int((a+b*ln(c*x^n))^2/x^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(4586\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}{\left (e x +d \right )^{2} x^{2}}\, dx \]________________________________________________________________________________________