\[ \int \frac {1}{x^2 \left (2-3 x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-3 x^{2}+2\right )^{\frac {1}{4}}}{2 x}+\frac {2^{\frac {3}{4}} \sqrt {\frac {\sqrt {-6 x^{2}+4}}{4}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {\arcsin \left (\frac {x \sqrt {6}}{2}\right )}{2}\right ), \sqrt {2}\right ) \sqrt {3}}{4 \cos \left (\frac {\arcsin \left (\frac {x \sqrt {6}}{2}\right )}{2}\right )} \]
command
int(1/x^2/(-3*x^2+2)^(3/4),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(-\frac {2^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {1}{2}\right ], \frac {3 x^{2}}{2}\right )}{2 x}\) | \(20\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} x^{2}}\, dx \]________________________________________________________________________________________