\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{10}} \, dx \]
Optimal antiderivative \[ \frac {4 a \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{9 b^{3}}-\frac {2 \left (a +\frac {b}{x^{3}}\right )^{\frac {5}{2}}}{15 b^{3}}-\frac {2 a^{2} \sqrt {a +\frac {b}{x^{3}}}}{3 b^{3}} \]
command
integrate(1/x^10/(a+b/x^3)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a^{2}}{3 \, b^{3}} - \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} - 10 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a\right )}}{45 \, b^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{10}}\,{d x} \]________________________________________________________________________________________