\[ \int \frac {\left (7+5 x^2\right )^2}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {x \left (-113 x^{2}+9\right )}{28 \sqrt {x^{4}+3 x^{2}+4}}-\frac {113 x \sqrt {x^{4}+3 x^{2}+4}}{28 \left (x^{2}+2\right )}+\frac {113 \left (x^{2}+2\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ), \frac {\sqrt {2}}{4}\right ) \sqrt {2}\, \sqrt {\frac {x^{4}+3 x^{2}+4}{\left (x^{2}+2\right )^{2}}}}{28 \cos \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ) \sqrt {x^{4}+3 x^{2}+4}}+\frac {9 \left (x^{2}+2\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ), \frac {\sqrt {2}}{4}\right ) \sqrt {\frac {x^{4}+3 x^{2}+4}{\left (x^{2}+2\right )^{2}}}\, \sqrt {2}}{8 \cos \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ) \sqrt {x^{4}+3 x^{2}+4}} \]
command
Integrate[(7 + 5*x^2)^2/(4 + 3*x^2 + x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {4 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (-9+113 x^2\right )+113 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )-\sqrt {2} \left (1043 i+113 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{112 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________