\[ \int \frac {x^8}{\left (1+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {x^{5}}{2 \sqrt {x^{4}+1}}+\frac {5 x \sqrt {x^{4}+1}}{6}-\frac {5 \left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{4}+1}{\left (x^{2}+1\right )^{2}}}}{12 \cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+1}} \]
command
integrate(x^8/(x^4+1)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {i} {\left (i \, x^{4} + i\right )} {\rm ellipticF}\left (\frac {\sqrt {i}}{x}, -1\right ) - {\left (2 \, x^{5} + 5 \, x\right )} \sqrt {x^{4} + 1}}{6 \, {\left (x^{4} + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1} x^{8}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]