\[ \int f^{a+b x^n} x^3 \, dx \]
Optimal antiderivative \[ -\frac {f^{a} x^{4} \Gamma \left (\frac {4}{n}, -b \,x^{n} \ln \left (f \right )\right ) \left (-b \,x^{n} \ln \left (f \right )\right )^{-\frac {4}{n}}}{n} \]
command
int(f^(a+b*x^n)*x^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {f^{a} \left (-b \right )^{-\frac {4}{n}} \ln \left (f \right )^{-\frac {4}{n}} \left (\frac {n \,x^{4} \left (-b \right )^{\frac {4}{n}} \ln \left (f \right )^{\frac {4}{n}} \left (\ln \left (f \right ) x^{n} b n +n +4\right ) \Gamma \left (1-\frac {4}{n}\right ) \Gamma \left (\frac {n +4}{n}+1\right ) L_{-\frac {4}{n}}^{\left (\frac {n +4}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{4 \left (n +4\right ) \Gamma \left (-\frac {4}{n}+\frac {n +4}{n}+1\right )}-\frac {n^{2} x^{n +4} \left (-b \right )^{\frac {4}{n}} \ln \left (f \right )^{1+\frac {4}{n}} b L_{-\frac {4}{n}}^{\left (\frac {n +4}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1-\frac {4}{n}\right ) \Gamma \left (\frac {n +4}{n}+1\right )}{4 \left (n +4\right ) \Gamma \left (-\frac {4}{n}+\frac {n +4}{n}+1\right )}\right )}{n}\) | \(212\) |
Maple 2021.1 output
\[ \int x^{3} f^{b \,x^{n}+a}\, dx \]________________________________________________________________________________________