\[ \int \left (7+5 x^2\right )^4 \sqrt {2+x^2-x^4} \, dx \]
Optimal antiderivative \[ -\frac {116100 x \left (-x^{4}+x^{2}+2\right )^{\frac {3}{2}}}{77}-\frac {14500 x^{3} \left (-x^{4}+x^{2}+2\right )^{\frac {3}{2}}}{33}-\frac {625 x^{5} \left (-x^{4}+x^{2}+2\right )^{\frac {3}{2}}}{11}+\frac {3764813 \EllipticE \left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{231}-\frac {539419 \EllipticF \left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{77}+\frac {x \left (717372 x^{2}+177953\right ) \sqrt {-x^{4}+x^{2}+2}}{231} \]
command
integrate((5*x^2+7)^4*(-x^4+x^2+2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (13125 \, x^{10} + 88375 \, x^{8} + 220550 \, x^{6} + 166072 \, x^{4} - 518647 \, x^{2} - 3764813\right )} \sqrt {-x^{4} + x^{2} + 2}}{231 \, x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt {-x^{4} + x^{2} + 2}, x\right ) \]