\[ \int \frac {e+f x}{(2+x) \sqrt {-1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e -2 f \right ) \arctan \left (\frac {\left (1-x \right )^{2}}{3 \sqrt {x^{3}-1}}\right )}{9}-\frac {2 \left (e +f \right ) \left (1-x \right ) \EllipticF \left (\frac {1-x +\sqrt {3}}{1-x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}+x +1}{\left (1-x -\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {x^{3}-1}\, \sqrt {\frac {-1+x}{\left (1-x -\sqrt {3}\right )^{2}}}} \]
command
integrate((f*x+e)/(2+x)/(x^3-1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{9} \, {\left (2 \, f - e\right )} \arctan \left (\frac {{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt {x^{3} - 1}}{6 \, {\left (x^{4} - x^{3} - x + 1\right )}}\right ) + \frac {2}{3} \, {\left (f + e\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{3} - 1} {\left (f x + e\right )}}{x^{4} + 2 \, x^{3} - x - 2}, x\right ) \]