\[ \int \frac {1}{\sqrt {3+6 x^2-2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (\frac {x \sqrt {-9+3 \sqrt {15}}}{3}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right ) \sqrt {18+6 \sqrt {15}}}{6} \]
command
integrate(1/(-2*x^4+6*x^2+3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{6} \, {\left (\sqrt {5} \sqrt {3} + 3\right )} \sqrt {\sqrt {5} \sqrt {3} - 3} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} \sqrt {\sqrt {5} \sqrt {3} - 3} x, -\sqrt {5} \sqrt {3} - 4\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 6 \, x^{2} + 3}}{2 \, x^{4} - 6 \, x^{2} - 3}, x\right ) \]