\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {2 a b \,d^{2} n x}{e^{3}}+\frac {2 b^{2} d^{2} n^{2} x}{e^{3}}-\frac {b^{2} d \,n^{2} x^{2}}{4 e^{2}}+\frac {2 b^{2} n^{2} x^{3}}{27 e}-\frac {2 b^{2} d^{2} n x \ln \left (c \,x^{n}\right )}{e^{3}}+\frac {b d n \,x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 e^{2}}-\frac {2 b n \,x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 e}+\frac {d^{2} x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{e^{3}}-\frac {d \,x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2 e^{2}}+\frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 e}-\frac {d^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{4}}-\frac {2 b \,d^{3} n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e x}{d}\right )}{e^{4}}+\frac {2 b^{2} d^{3} n^{2} \polylog \left (3, -\frac {e x}{d}\right )}{e^{4}} \]
command
int(x^3*(a+b*ln(c*x^n))^2/(e*x+d),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(4508\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{3}}{e x +d}\, dx \]________________________________________________________________________________________