\[ \int \left (7+5 x^2\right )^4 \left (2+x^2-x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {x \left (-1581440 x^{2}+69817\right ) \left (-x^{4}+x^{2}+2\right )^{\frac {3}{2}}}{1001}-\frac {132300 x \left (-x^{4}+x^{2}+2\right )^{\frac {5}{2}}}{143}-\frac {11750 x^{3} \left (-x^{4}+x^{2}+2\right )^{\frac {5}{2}}}{39}-\frac {125 x^{5} \left (-x^{4}+x^{2}+2\right )^{\frac {5}{2}}}{3}+\frac {124141422 \EllipticE \left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{5005}-\frac {50794416 \EllipticF \left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{5005}+\frac {3 x \left (7837383 x^{2}+2193559\right ) \sqrt {-x^{4}+x^{2}+2}}{5005} \]
command
integrate((5*x^2+7)^4*(-x^4+x^2+2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (625625 \, x^{14} + 3272500 \, x^{12} + 2967125 \, x^{10} - 15130150 \, x^{8} - 45845855 \, x^{6} - 43271392 \, x^{4} + 37918479 \, x^{2} + 372424266\right )} \sqrt {-x^{4} + x^{2} + 2}}{15015 \, x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (625 \, x^{12} + 2875 \, x^{10} + 2600 \, x^{8} - 7490 \, x^{6} - 19159 \, x^{4} - 16121 \, x^{2} - 4802\right )} \sqrt {-x^{4} + x^{2} + 2}, x\right ) \]