\[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {-x^{3}-1}}{5 x^{5}}-\frac {7 \sqrt {-x^{3}-1}}{20 x^{2}}+\frac {7 \left (1+x \right ) \EllipticF \left (\frac {1+x +\sqrt {3}}{1+x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x -\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{60 \sqrt {-x^{3}-1}\, \sqrt {\frac {-1-x}{\left (1+x -\sqrt {3}\right )^{2}}}} \]
command
integrate(1/x^6/(-x^3-1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (7 \, x^{3} - 4\right )} \sqrt {-x^{3} - 1}}{20 \, x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-x^{3} - 1}}{x^{9} + x^{6}}, x\right ) \]