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