\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 x \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{9}+\frac {4 \,2^{\frac {3}{4}} \sqrt {\frac {\sqrt {-6 x^{2}+4}}{4}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {\arcsin \left (\frac {x \sqrt {6}}{2}\right )}{2}\right ), \sqrt {2}\right ) \sqrt {3}}{27 \cos \left (\frac {\arcsin \left (\frac {x \sqrt {6}}{2}\right )}{2}\right )} \]
command
int(x^2/(-3*x^2+2)^(3/4),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {2^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], \frac {3 x^{2}}{2}\right )}{6}\) | \(20\) |
Maple 2021.1 output
\[ \int \frac {x^{2}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}}}\, dx \]________________________________________________________________________________________