\[ \int \frac {1}{x^5 \left (2+x^6\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{6 x^{4} \sqrt {x^{6}+2}}-\frac {7 \sqrt {x^{6}+2}}{48 x^{4}}-\frac {7 \,2^{\frac {5}{6}} \left (2^{\frac {1}{3}}+x^{2}\right ) \EllipticF \left (\frac {x^{2}+2^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{x^{2}+2^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {2^{\frac {2}{3}}-2^{\frac {1}{3}} x^{2}+x^{4}}{\left (x^{2}+2^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{288 \sqrt {x^{6}+2}\, \sqrt {\frac {2^{\frac {1}{3}}+x^{2}}{\left (x^{2}+2^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(1/x^5/(x^6+2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {7 \, {\left (x^{10} + 2 \, x^{4}\right )} {\rm weierstrassPInverse}\left (0, -8, x^{2}\right ) + {\left (7 \, x^{6} + 6\right )} \sqrt {x^{6} + 2}}{48 \, {\left (x^{10} + 2 \, x^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{6} + 2}}{x^{17} + 4 \, x^{11} + 4 \, x^{5}}, x\right ) \]