\[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {20 x^{2} \sqrt {x^{3}+1}}{91}+\frac {2 x^{5} \sqrt {x^{3}+1}}{13}+\frac {80 \sqrt {x^{3}+1}}{91 \left (1+x +\sqrt {3}\right )}+\frac {80 \left (1+x \right ) \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{273 \sqrt {x^{3}+1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}}-\frac {40 \,3^{\frac {1}{4}} \left (1+x \right ) \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{91 \sqrt {x^{3}+1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate(x^7/(x^3+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{91} \, {\left (7 \, x^{5} - 10 \, x^{2}\right )} \sqrt {x^{3} + 1} - \frac {80}{91} \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {x^{7}}{\sqrt {x^{3} + 1}}, x\right ) \]