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