\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {2 a b n x}{e^{4}}+\frac {2 b^{2} n^{2} x}{e^{4}}-\frac {b^{2} d^{2} n^{2}}{3 e^{5} \left (e x +d \right )}-\frac {b^{2} d \,n^{2} \ln \left (x \right )}{3 e^{5}}-\frac {2 b^{2} n x \ln \left (c \,x^{n}\right )}{e^{4}}+\frac {b \,d^{3} n \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 e^{5} \left (e x +d \right )^{2}}+\frac {10 b d n x \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 e^{4} \left (e x +d \right )}-\frac {5 d \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 e^{5}}+\frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{e^{4}}-\frac {d^{4} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 e^{5} \left (e x +d \right )^{3}}+\frac {2 d^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{e^{5} \left (e x +d \right )^{2}}+\frac {6 d x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{e^{4} \left (e x +d \right )}-\frac {3 b^{2} d \,n^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {26 b d n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{3 e^{5}}-\frac {4 d \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{5}}-\frac {26 b^{2} d \,n^{2} \polylog \left (2, -\frac {e x}{d}\right )}{3 e^{5}}-\frac {8 b d n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e x}{d}\right )}{e^{5}}+\frac {8 b^{2} d \,n^{2} \polylog \left (3, -\frac {e x}{d}\right )}{e^{5}} \]
command
int(x^4*(a+b*ln(c*x^n))^2/(e*x+d)^4,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(6114\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{4}}{\left (e x +d \right )^{4}}\, dx \]________________________________________________________________________________________