\[ \int \frac {1}{x^2 \left (-2-3 x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 x^{2}-2\right )^{\frac {1}{4}}}{2 x}+\frac {2^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\left (-24 x^{2}-16\right )^{\frac {1}{4}}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {\left (-3 x^{2}-2\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {2}+\sqrt {-3 x^{2}-2}\right ) \sqrt {-\frac {x^{2}}{\left (\sqrt {2}+\sqrt {-3 x^{2}-2}\right )^{2}}}\, \sqrt {3}}{8 \cos \left (2 \arctan \left (\frac {\left (-3 x^{2}-2\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right ) x} \]
command
int(1/x^2/(-3*x^2-2)^(3/4),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| meijerg | \(\frac {\left (-1\right )^{\frac {1}{4}} 2^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {1}{2}\right ], -\frac {3 x^{2}}{2}\right )}{2 x}\) | \(23\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (-3 x^{2}-2\right )^{\frac {3}{4}} x^{2}}\, dx \]________________________________________________________________________________________