\[ \int f^{a+b x^n} g^{c+d x^n} \, dx \]
Optimal antiderivative \[ -\frac {f^{a} g^{c} x \Gamma \left (\frac {1}{n}, -x^{n} \left (b \ln \left (f \right )+d \ln \left (g \right )\right )\right ) \left (-x^{n} \left (b \ln \left (f \right )+d \ln \left (g \right )\right )\right )^{-\frac {1}{n}}}{n} \]
command
int(f^(a+b*x^n)*g^(c+d*x^n),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {f^{a} g^{c} \left (-d \right )^{-\frac {1}{n}} \ln \left (g \right )^{-\frac {1}{n}} \left (1+\frac {b \ln \left (f \right )}{d \ln \left (g \right )}\right )^{-\frac {1}{n}} \left (\frac {n x \left (-d \right )^{\frac {1}{n}} \ln \left (g \right )^{\frac {1}{n}} \left (1+\frac {b \ln \left (f \right )}{d \ln \left (g \right )}\right )^{\frac {1}{n}} \left (x^{n} d \ln \left (g \right ) \left (1+\frac {b \ln \left (f \right )}{d \ln \left (g \right )}\right ) n +n +1\right ) \Gamma \left (1-\frac {1}{n}\right ) \Gamma \left (\frac {1+n}{n}+1\right ) L_{-\frac {1}{n}}^{\left (\frac {1+n}{n}\right )}\left (x^{n} d \ln \left (g \right ) \left (1+\frac {b \ln \left (f \right )}{d \ln \left (g \right )}\right )\right )}{\left (1+n \right ) \Gamma \left (-\frac {1}{n}+\frac {1+n}{n}+1\right )}-\frac {n^{2} x^{1+n} \left (-d \right )^{\frac {1}{n}} \ln \left (g \right )^{1+\frac {1}{n}} \left (1+\frac {b \ln \left (f \right )}{d \ln \left (g \right )}\right )^{1+\frac {1}{n}} d L_{-\frac {1}{n}}^{\left (\frac {1+n}{n}+1\right )}\left (x^{n} d \ln \left (g \right ) \left (1+\frac {b \ln \left (f \right )}{d \ln \left (g \right )}\right )\right ) \Gamma \left (1-\frac {1}{n}\right ) \Gamma \left (\frac {1+n}{n}+1\right )}{\left (1+n \right ) \Gamma \left (-\frac {1}{n}+\frac {1+n}{n}+1\right )}\right )}{n}\) | \(298\) |
Maple 2021.1 output
\[ \int f^{b \,x^{n}+a} g^{d \,x^{n}+c}\, dx \]________________________________________________________________________________________