\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {2 b^{2} n^{2}}{d^{4} x}-\frac {b^{2} e \,n^{2}}{3 d^{4} \left (e x +d \right )}-\frac {b^{2} e \,n^{2} \ln \left (x \right )}{3 d^{5}}-\frac {2 b n \left (a +b \ln \left (c \,x^{n}\right )\right )}{d^{4} x}+\frac {b e n \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 d^{3} \left (e x +d \right )^{2}}-\frac {8 b \,e^{2} n x \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 d^{5} \left (e x +d \right )}+\frac {4 e \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 d^{5}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{d^{4} x}-\frac {e \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{3 d^{2} \left (e x +d \right )^{3}}-\frac {e \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{d^{3} \left (e x +d \right )^{2}}+\frac {3 e^{2} x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{d^{5} \left (e x +d \right )}-\frac {4 e \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{3 b \,d^{5} n}+\frac {3 b^{2} e \,n^{2} \ln \left (e x +d \right )}{d^{5}}-\frac {26 b e n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{3 d^{5}}+\frac {4 e \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{5}}-\frac {26 b^{2} e \,n^{2} \polylog \left (2, -\frac {e x}{d}\right )}{3 d^{5}}+\frac {8 b e n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e x}{d}\right )}{d^{5}}-\frac {8 b^{2} e \,n^{2} \polylog \left (3, -\frac {e x}{d}\right )}{d^{5}} \]
command
int((a+b*ln(c*x^n))^2/x^2/(e*x+d)^4,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(6791\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}{\left (e x +d \right )^{4} x^{2}}\, dx \]________________________________________________________________________________________