\[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {\cos \left (4 \arctan \left (2^{\frac {1}{4}} x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (2^{\frac {1}{4}} x \right )\right ), \frac {\sqrt {8-5 \sqrt {2}}}{4}\right ) \left (1+x^{2} \sqrt {2}\right ) \sqrt {\frac {4 x^{4}+5 x^{2}+2}{\left (1+x^{2} \sqrt {2}\right )^{2}}}\, 2^{\frac {1}{4}}}{4 \cos \left (2 \arctan \left (2^{\frac {1}{4}} x \right )\right ) \sqrt {4 x^{4}+5 x^{2}+2}} \]
command
integrate(1/(4*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
\[ -\frac {1}{32} \, \sqrt {2} {\left (\sqrt {-7} + 5\right )} \sqrt {\sqrt {-7} - 5} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {-7} - 5}, \frac {5}{16} \, \sqrt {-7} + \frac {9}{16}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {4 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]