\[ \int \frac {1}{\sqrt {3+9 x^2+2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {1}{36+x^{2} \left (54+6 \sqrt {57}\right )}}\, \sqrt {36+x^{2} \left (54+6 \sqrt {57}\right )}\, \EllipticF \left (\frac {x \sqrt {54+6 \sqrt {57}}}{\sqrt {36+x^{2} \left (54+6 \sqrt {57}\right )}}, \frac {\sqrt {-19+3 \sqrt {57}}}{2}\right ) \left (6+x^{2} \left (\sqrt {57}+9\right )\right ) \sqrt {\frac {6+x^{2} \left (9-\sqrt {57}\right )}{6+x^{2} \left (\sqrt {57}+9\right )}}}{\sqrt {2 x^{4}+9 x^{2}+3}\, \sqrt {54+6 \sqrt {57}}} \]
command
integrate(1/(2*x^4+9*x^2+3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{24} \, {\left (\sqrt {19} \sqrt {6} + 3 \, \sqrt {6} \sqrt {3}\right )} \sqrt {\sqrt {19} \sqrt {3} - 9} {\rm ellipticF}\left (\frac {1}{6} \, \sqrt {6} \sqrt {\sqrt {19} \sqrt {3} - 9} x, \frac {3}{4} \, \sqrt {19} \sqrt {3} + \frac {23}{4}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 9 \, x^{2} + 3}}, x\right ) \]