\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {7679 x \left (x^{2}+2\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {x \left (179 x^{2}+115\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {7679 \left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {\frac {1}{x^{2}+1}}\, \EllipticE \left (\frac {x}{\sqrt {x^{2}+1}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {x^{2}+2}{x^{2}+1}}}{2 \sqrt {x^{4}+3 x^{2}+2}}+\frac {15383 \left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {\frac {1}{x^{2}+1}}\, \EllipticF \left (\frac {x}{\sqrt {x^{2}+1}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {x^{2}+2}{x^{2}+1}}}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5000 x \sqrt {x^{4}+3 x^{2}+2}}{3}+625 x^{3} \sqrt {x^{4}+3 x^{2}+2} \]
command
Integrate[(7 + 5*x^2)^5/(2 + 3*x^2 + x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {19655 x+36963 x^3+21250 x^5+3750 x^7-23037 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-7729 i \sqrt {1+x^2} \sqrt {2+x^2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )}{6 \sqrt {2+3 x^2+x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________