\[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\left (1+x \right ) \sqrt {3+2 \sqrt {3}}}{\sqrt {x^{3}+1}}\right ) \sqrt {2}\, 3^{\frac {1}{4}}}{3}+\frac {\left (1+x \right ) \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {1}{4}}}{3 \sqrt {x^{3}+1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate(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
\[ -\frac {1}{3} \, {\left (\sqrt {3} - 3\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) - \frac {1}{6} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (-\frac {3^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} - \sqrt {3} {\left (x^{2} + 2 \, x + 4\right )} - 6 \, x\right )}}{12 \, \sqrt {x^{3} + 1}}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{3} + 1} {\left (x^{2} - \sqrt {3} x + x\right )}}{x^{5} + 2 \, x^{4} - 2 \, x^{3} + x^{2} + 2 \, x - 2}, x\right ) \]