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