\[ \int f^{a+b x^n} x \, dx \]
Optimal antiderivative \[ -\frac {f^{a} x^{2} \Gamma \left (\frac {2}{n}, -b \,x^{n} \ln \left (f \right )\right ) \left (-b \,x^{n} \ln \left (f \right )\right )^{-\frac {2}{n}}}{n} \]
command
int(f^(a+b*x^n)*x,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {f^{a} \left (-b \right )^{-\frac {2}{n}} \ln \left (f \right )^{-\frac {2}{n}} \left (\frac {n \,x^{2} \left (-b \right )^{\frac {2}{n}} \ln \left (f \right )^{\frac {2}{n}} \left (\ln \left (f \right ) x^{n} b n +n +2\right ) \Gamma \left (1-\frac {2}{n}\right ) \Gamma \left (\frac {2+n}{n}+1\right ) L_{-\frac {2}{n}}^{\left (\frac {2+n}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{2 \left (2+n \right ) \Gamma \left (-\frac {2}{n}+\frac {2+n}{n}+1\right )}-\frac {n^{2} x^{2+n} \left (-b \right )^{\frac {2}{n}} \ln \left (f \right )^{1+\frac {2}{n}} b L_{-\frac {2}{n}}^{\left (\frac {2+n}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1-\frac {2}{n}\right ) \Gamma \left (\frac {2+n}{n}+1\right )}{2 \left (2+n \right ) \Gamma \left (-\frac {2}{n}+\frac {2+n}{n}+1\right )}\right )}{n}\) | \(212\) |
Maple 2021.1 output
\[ \int x \,f^{b \,x^{n}+a}\, dx \]________________________________________________________________________________________