\[ \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {x \left (65345 x^{2}+208212\right ) \left (x^{4}+3 x^{2}+2\right )^{\frac {3}{2}}}{3003}+\frac {3825 x \left (x^{4}+3 x^{2}+2\right )^{\frac {5}{2}}}{143}+\frac {125 x^{3} \left (x^{4}+3 x^{2}+2\right )^{\frac {5}{2}}}{13}+\frac {20884 x \left (x^{2}+2\right )}{65 \sqrt {x^{4}+3 x^{2}+2}}-\frac {20884 \left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {\frac {1}{x^{2}+1}}\, \EllipticE \left (\frac {x}{\sqrt {x^{2}+1}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {x^{2}+2}{x^{2}+1}}}{65 \sqrt {x^{4}+3 x^{2}+2}}+\frac {1171349 \left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {\frac {1}{x^{2}+1}}\, \EllipticF \left (\frac {x}{\sqrt {x^{2}+1}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {x^{2}+2}{x^{2}+1}}}{5005 \sqrt {x^{4}+3 x^{2}+2}}+\frac {x \left (297911 x^{2}+1032541\right ) \sqrt {x^{4}+3 x^{2}+2}}{5005} \]
command
Integrate[(7 + 5*x^2)^3*(2 + 3*x^2 + x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {13572486 x+40493455 x^3+54938052 x^5+46218643 x^7+25350660 x^9+8705725 x^{11}+1701000 x^{13}+144375 x^{15}-4824204 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-2203890 i \sqrt {1+x^2} \sqrt {2+x^2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )}{15015 \sqrt {2+3 x^2+x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________