\[ \int \frac {1}{\sqrt {2+5 x^2+x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {1}{4+x^{2} \left (5+\sqrt {17}\right )}}\, \left (4+x^{2} \left (5+\sqrt {17}\right )\right )^{\frac {3}{2}} \EllipticF \left (\frac {x \sqrt {5+\sqrt {17}}}{\sqrt {4+x^{2} \left (5+\sqrt {17}\right )}}, \frac {\sqrt {-17+5 \sqrt {17}}}{2}\right ) \sqrt {\frac {4+x^{2} \left (5-\sqrt {17}\right )}{4+x^{2} \left (5+\sqrt {17}\right )}}}{2 \sqrt {x^{4}+5 x^{2}+2}\, \sqrt {5+\sqrt {17}}} \]
command
integrate(1/(x^4+5*x^2+2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{8} \, {\left (\sqrt {17} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {17} - 5} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {17} - 5}, \frac {5}{4} \, \sqrt {17} + \frac {21}{4}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}, x\right ) \]