\[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (\frac {4 x}{\sqrt {5+\sqrt {89}}}, \frac {5 i}{8}+\frac {i \sqrt {89}}{8}\right ) \sqrt {2}}{\sqrt {-5+\sqrt {89}}} \]
command
integrate(1/(-8*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
\[ \frac {1}{64} \, {\left (\sqrt {89} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {89} - 5} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {89} - 5}, -\frac {5}{32} \, \sqrt {89} - \frac {57}{32}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-8 \, x^{4} + 5 \, x^{2} + 2}}{8 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \]