\[ \int \frac {x^8}{\sqrt {1+x^4}} \, dx \]
Optimal antiderivative \[ -\frac {5 x \sqrt {x^{4}+1}}{21}+\frac {x^{5} \sqrt {x^{4}+1}}{7}+\frac {5 \left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{4}+1}{\left (x^{2}+1\right )^{2}}}}{42 \cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+1}} \]
command
integrate(x^8/(x^4+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{21} \, {\left (3 \, x^{5} - 5 \, x\right )} \sqrt {x^{4} + 1} + \frac {5}{21} i \, \sqrt {i} {\rm ellipticF}\left (\frac {\sqrt {i}}{x}, -1\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {x^{8}}{\sqrt {x^{4} + 1}}, x\right ) \]