8.46 Problem number 2021

\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^7} \, dx \]

Optimal antiderivative \[ -\frac {2 \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{9 b^{2}}+\frac {2 a \sqrt {a +\frac {b}{x^{3}}}}{3 b^{2}} \]

command

integrate(1/x^7/(a+b/x^3)^(1/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b^{2}} + \frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a}{3 \, b^{2}} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \int \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{7}}\,{d x} \]________________________________________________________________________________________