\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{13}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{3 b^{4}}+\frac {2 a \left (a +\frac {b}{x^{3}}\right )^{\frac {5}{2}}}{5 b^{4}}-\frac {2 \left (a +\frac {b}{x^{3}}\right )^{\frac {7}{2}}}{21 b^{4}}+\frac {2 a^{3} \sqrt {a +\frac {b}{x^{3}}}}{3 b^{4}} \]
command
integrate(1/x^13/(a+b/x^3)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a^{3}}{3 \, b^{4}} - \frac {2 \, {\left (5 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} - 21 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a + 35 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, b^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{13}}\,{d x} \]________________________________________________________________________________________