\[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {x \left (35 x^{2}+108\right ) \left (x^{4}+3 x^{2}+4\right )^{\frac {3}{2}}}{63}+\frac {2798 x \sqrt {x^{4}+3 x^{2}+4}}{105 \left (x^{2}+2\right )}+\frac {x \left (289 x^{2}+1029\right ) \sqrt {x^{4}+3 x^{2}+4}}{105}-\frac {2798 \left (x^{2}+2\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ), \frac {\sqrt {2}}{4}\right ) \sqrt {2}\, \sqrt {\frac {x^{4}+3 x^{2}+4}{\left (x^{2}+2\right )^{2}}}}{105 \cos \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ) \sqrt {x^{4}+3 x^{2}+4}}+\frac {74 \left (x^{2}+2\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ), \frac {\sqrt {2}}{4}\right ) \sqrt {\frac {x^{4}+3 x^{2}+4}{\left (x^{2}+2\right )^{2}}}\, \sqrt {2}}{3 \cos \left (2 \arctan \left (\frac {x \sqrt {2}}{2}\right )\right ) \sqrt {x^{4}+3 x^{2}+4}} \]
command
Integrate[(7 + 5*x^2)*(4 + 3*x^2 + x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (20988+28489 x^2+19068 x^4+7082 x^6+1590 x^8+175 x^{10}\right )-4197 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+3 \sqrt {2} \left (-567 i+1399 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{630 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________