\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx \]
Optimal antiderivative \[ \frac {3 b e n x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2 d^{3} \left (e x +d \right )}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 d^{3}}+\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 d \left (e x +d \right )^{2}}-\frac {e x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{d^{3} \left (e x +d \right )}+\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{4}}{4 b \,d^{3} n}-\frac {3 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{d^{3}}+\frac {9 b n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {e x}{d}\right )}{2 d^{3}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (1+\frac {e x}{d}\right )}{d^{3}}-\frac {3 b^{3} n^{3} \polylog \left (2, -\frac {e x}{d}\right )}{d^{3}}+\frac {9 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e x}{d}\right )}{d^{3}}-\frac {3 b n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (2, -\frac {e x}{d}\right )}{d^{3}}-\frac {9 b^{3} n^{3} \polylog \left (3, -\frac {e x}{d}\right )}{d^{3}}+\frac {6 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, -\frac {e x}{d}\right )}{d^{3}}-\frac {6 b^{3} n^{3} \polylog \left (4, -\frac {e x}{d}\right )}{d^{3}} \]
command
int((a+b*ln(c*x^n))^3/x/(e*x+d)^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(19018\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3}}{\left (e x +d \right )^{3} x}\, dx \]________________________________________________________________________________________