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