\[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {15 \EllipticF \left (x , i\right )}{14}+\frac {1}{2 x^{7} \sqrt {-x^{4}+1}}-\frac {9 \sqrt {-x^{4}+1}}{14 x^{7}}-\frac {15 \sqrt {-x^{4}+1}}{14 x^{3}} \]
command
integrate(1/x^8/(-x^4+1)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {15 \, {\left (x^{11} - x^{7}\right )} {\rm ellipticF}\left (x, -1\right ) - {\left (15 \, x^{8} - 6 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{14 \, {\left (x^{11} - x^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1}}{x^{16} - 2 \, x^{12} + x^{8}}, x\right ) \]