\[ \int \frac {\sqrt {a x^{2 n}}}{\sqrt {1+x^n}} \, dx \]
Optimal antiderivative \[ \frac {x \hypergeom \left (\left [\frac {1}{2}, 1+\frac {1}{n}\right ], \left [2+\frac {1}{n}\right ], -x^{n}\right ) \sqrt {a \,x^{2 n}}}{1+n} \]
command
int((a*x^(2*n))^(1/2)/(1+x^n)^(1/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {x \hypergeom \left (\left [\frac {1}{2}, 1+\frac {1}{n}\right ], \left [2+\frac {1}{n}\right ], -x^{n}\right ) \sqrt {a \,x^{2 n}}}{1+n}\) | \(36\) |
Maple 2021.1 output
\[ \int \frac {\sqrt {a \,x^{2 n}}}{\sqrt {x^{n}+1}}\, dx \]________________________________________________________________________________________