\[ \int f^{a+b x^n} x^m \, dx \]
Optimal antiderivative \[ -\frac {f^{a} x^{1+m} \Gamma \left (\frac {1+m}{n}, -b \,x^{n} \ln \left (f \right )\right ) \left (-b \,x^{n} \ln \left (f \right )\right )^{-\frac {1+m}{n}}}{n} \]
command
int(f^(a+b*x^n)*x^m,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {f^{a} \left (-b \right )^{-\frac {m}{n}-\frac {1}{n}} \ln \left (f \right )^{-\frac {m}{n}-\frac {1}{n}} \left (\frac {n \,x^{1+m} \left (-b \right )^{\frac {m}{n}+\frac {1}{n}} \ln \left (f \right )^{\frac {m}{n}+\frac {1}{n}} \left (\ln \left (f \right ) x^{n} b n +m +n +1\right ) L_{-\frac {1+m}{n}}^{\left (\frac {1+m +n}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (-\frac {1+m}{n}+1\right ) \Gamma \left (\frac {1+m +n}{n}+1\right )}{\left (1+m \right ) \left (1+m +n \right ) \Gamma \left (-\frac {1+m}{n}+\frac {1+m +n}{n}+1\right )}-\frac {n^{2} x^{1+m +n} \left (-b \right )^{\frac {m}{n}+\frac {1}{n}} \ln \left (f \right )^{1+\frac {m}{n}+\frac {1}{n}} b L_{-\frac {1+m}{n}}^{\left (\frac {1+m +n}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (-\frac {1+m}{n}+1\right ) \Gamma \left (\frac {1+m +n}{n}+1\right )}{\left (1+m \right ) \left (1+m +n \right ) \Gamma \left (-\frac {1+m}{n}+\frac {1+m +n}{n}+1\right )}\right )}{n}\) | \(280\) |
Maple 2021.1 output
\[ \int f^{b \,x^{n}+a} x^{m}\, dx \]________________________________________________________________________________________