\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {a +\frac {b}{x^{3}}}}{19 b \,x^{8}}+\frac {32 a \sqrt {a +\frac {b}{x^{3}}}}{247 b^{2} x^{5}}-\frac {320 a^{2} \sqrt {a +\frac {b}{x^{3}}}}{1729 b^{3} x^{2}}+\frac {1280 a^{3} \sqrt {a +\frac {b}{x^{3}}}}{1729 b^{\frac {11}{3}} \left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {1280 a^{\frac {10}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right ) \EllipticF \left (\frac {\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {a^{\frac {2}{3}}+\frac {b^{\frac {2}{3}}}{x^{2}}-\frac {a^{\frac {1}{3}} b^{\frac {1}{3}}}{x}}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{5187 b^{\frac {11}{3}} \sqrt {a +\frac {b}{x^{3}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right )}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {640 \,3^{\frac {1}{4}} a^{\frac {10}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right ) \EllipticE \left (\frac {\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}+\frac {b^{\frac {2}{3}}}{x^{2}}-\frac {a^{\frac {1}{3}} b^{\frac {1}{3}}}{x}}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{1729 b^{\frac {11}{3}} \sqrt {a +\frac {b}{x^{3}}}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+\frac {b^{\frac {1}{3}}}{x}\right )}{\left (\frac {b^{\frac {1}{3}}}{x}+a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(1/x^12/(a+b/x^3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (640 \, a^{3} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right )\right ) + {\left (160 \, a^{2} b x^{6} - 112 \, a b^{2} x^{3} + 91 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{1729 \, b^{4} x^{8}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\frac {a x^{3} + b}{x^{3}}}}{a x^{12} + b x^{9}}, x\right ) \]