\[ \int \frac {f^{a+b x^n}}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {f^{a} \Gamma \left (-\frac {1}{n}, -b \,x^{n} \ln \left (f \right )\right ) \left (-b \,x^{n} \ln \left (f \right )\right )^{\frac {1}{n}}}{n x} \]
command
int(f^(a+b*x^n)/x^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {f^{a} \left (-b \right )^{\frac {1}{n}} \ln \left (f \right )^{\frac {1}{n}} \left (-\frac {n \left (-b \right )^{-\frac {1}{n}} \ln \left (f \right )^{-\frac {1}{n}} \left (\ln \left (f \right ) x^{n} b n +n -1\right ) \Gamma \left (1+\frac {1}{n}\right ) \Gamma \left (\frac {-1+n}{n}+1\right ) L_{\frac {1}{n}}^{\left (\frac {-1+n}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{x \left (-1+n \right ) \Gamma \left (\frac {1}{n}+\frac {-1+n}{n}+1\right )}+\frac {n^{2} x^{-1+n} \left (-b \right )^{-\frac {1}{n}} \ln \left (f \right )^{1-\frac {1}{n}} b L_{\frac {1}{n}}^{\left (\frac {-1+n}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1+\frac {1}{n}\right ) \Gamma \left (\frac {-1+n}{n}+1\right )}{\left (-1+n \right ) \Gamma \left (\frac {1}{n}+\frac {-1+n}{n}+1\right )}\right )}{n}\) | \(195\) |
Maple 2021.1 output
\[ \int \frac {f^{b \,x^{n}+a}}{x^{2}}\, dx \]________________________________________________________________________________________