\[ \int \frac {2+3 x^2}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {x \left (8 x^{2}+7\right )}{39 \sqrt {x^{4}+5 x^{2}+3}}+\frac {4 x \left (5+2 x^{2}+\sqrt {13}\right )}{39 \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {\frac {1}{36+x^{2} \left (30+6 \sqrt {13}\right )}}\, \sqrt {36+x^{2} \left (30+6 \sqrt {13}\right )}\, \EllipticE \left (\frac {x \sqrt {30+6 \sqrt {13}}}{\sqrt {36+x^{2} \left (30+6 \sqrt {13}\right )}}, \frac {\sqrt {-78+30 \sqrt {13}}}{6}\right ) \left (6+x^{2} \left (5+\sqrt {13}\right )\right ) \sqrt {30+6 \sqrt {13}}\, \sqrt {\frac {6+x^{2} \left (5-\sqrt {13}\right )}{6+x^{2} \left (5+\sqrt {13}\right )}}}{117 \sqrt {x^{4}+5 x^{2}+3}}+\frac {11 \sqrt {\frac {1}{36+x^{2} \left (30+6 \sqrt {13}\right )}}\, \sqrt {36+x^{2} \left (30+6 \sqrt {13}\right )}\, \EllipticF \left (\frac {x \sqrt {30+6 \sqrt {13}}}{\sqrt {36+x^{2} \left (30+6 \sqrt {13}\right )}}, \frac {\sqrt {-78+30 \sqrt {13}}}{6}\right ) \left (6+x^{2} \left (5+\sqrt {13}\right )\right ) \sqrt {\frac {6+x^{2} \left (5-\sqrt {13}\right )}{6+x^{2} \left (5+\sqrt {13}\right )}}}{13 \sqrt {x^{4}+5 x^{2}+3}\, \sqrt {30+6 \sqrt {13}}} \]
command
Integrate[(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {-2 x \left (7+8 x^2\right )+4 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (13+4 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{78 \sqrt {3+5 x^2+x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________