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