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