\[ \int \frac {1}{\sqrt {-2-x^2-3 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {\cos \left (4 \arctan \left (\frac {x 24^{\frac {1}{4}}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {3^{\frac {1}{4}} 2^{\frac {3}{4}} x}{2}\right )\right ), \frac {\sqrt {72-6 \sqrt {6}}}{12}\right ) \left (2+x^{2} \sqrt {6}\right ) \sqrt {\frac {3 x^{4}+x^{2}+2}{\left (2+x^{2} \sqrt {6}\right )^{2}}}\, 6^{\frac {3}{4}}}{12 \cos \left (2 \arctan \left (\frac {3^{\frac {1}{4}} 2^{\frac {3}{4}} x}{2}\right )\right ) \sqrt {-3 x^{4}-x^{2}-2}} \]
command
integrate(1/(-3*x^4-x^2-2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{24} \, \sqrt {-2} {\left (\sqrt {-23} + 1\right )} \sqrt {\sqrt {-23} - 1} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {-23} - 1}, \frac {1}{12} \, \sqrt {-23} - \frac {11}{12}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} - x^{2} - 2}}{3 \, x^{4} + x^{2} + 2}, x\right ) \]