\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {a +\frac {b}{x^{3}}}}{13 x^{5}}-\frac {6 a \sqrt {a +\frac {b}{x^{3}}}}{91 b \,x^{2}}+\frac {24 a^{2} \sqrt {a +\frac {b}{x^{3}}}}{91 b^{\frac {5}{3}} \left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {8 \,3^{\frac {3}{4}} a^{\frac {7}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right ) \EllipticF \left (\frac {\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {a^{\frac {2}{3}}+\frac {b^{\frac {2}{3}}}{x^{2}}-\frac {a^{\frac {1}{3}} b^{\frac {1}{3}}}{x}}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{91 b^{\frac {5}{3}} \sqrt {a +\frac {b}{x^{3}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right )}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {12 \,3^{\frac {1}{4}} a^{\frac {7}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right ) \EllipticE \left (\frac {\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}+\frac {b^{\frac {2}{3}}}{x^{2}}-\frac {a^{\frac {1}{3}} b^{\frac {1}{3}}}{x}}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{91 b^{\frac {5}{3}} \sqrt {a +\frac {b}{x^{3}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right )}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((a+b/x^3)^(1/2)/x^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, a^{2} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right )\right ) + {\left (3 \, a b x^{3} + 7 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{91 \, b^{2} x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\frac {a x^{3} + b}{x^{3}}}}{x^{6}}, x\right ) \]