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