\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^8} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {a +\frac {b}{x^{3}}}}{17 x^{7}}-\frac {6 a \sqrt {a +\frac {b}{x^{3}}}}{187 b \,x^{4}}+\frac {48 a^{2} \sqrt {a +\frac {b}{x^{3}}}}{935 b^{2} x}-\frac {32 \,3^{\frac {3}{4}} a^{3} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right ) \EllipticF \left (\frac {\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}+\frac {b^{\frac {2}{3}}}{x^{2}}-\frac {a^{\frac {1}{3}} b^{\frac {1}{3}}}{x}}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{935 b^{\frac {7}{3}} \sqrt {a +\frac {b}{x^{3}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right )}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((a+b/x^3)^(1/2)/x^8,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (48 \, a^{3} \sqrt {b} x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right ) - {\left (24 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3} - 55 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{935 \, b^{3} x^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\frac {a x^{3} + b}{x^{3}}}}{x^{8}}, x\right ) \]