\[ \int \frac {1}{x^2 \sqrt {-1-x^3}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {-x^{3}-1}}{x}-\frac {\sqrt {-x^{3}-1}}{1+x -\sqrt {3}}-\frac {\left (1+x \right ) \EllipticF \left (\frac {1+x +\sqrt {3}}{1+x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \sqrt {2}\, \sqrt {\frac {x^{2}-x +1}{\left (1+x -\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{3 \sqrt {-x^{3}-1}\, \sqrt {\frac {-1-x}{\left (1+x -\sqrt {3}\right )^{2}}}}+\frac {3^{\frac {1}{4}} \left (1+x \right ) \EllipticE \left (\frac {1+x +\sqrt {3}}{1+x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x -\sqrt {3}\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{2 \sqrt {-x^{3}-1}\, \sqrt {\frac {-1-x}{\left (1+x -\sqrt {3}\right )^{2}}}} \]
command
integrate(1/x^2/(-x^3-1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {-x^{3} - 1}}{x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-x^{3} - 1}}{x^{5} + x^{2}}, x\right ) \]