\[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx \]
Optimal antiderivative \[ -\frac {1}{6 a \left (a +\frac {b}{x^{4}}\right )^{\frac {3}{2}} x}-\frac {5}{12 a^{2} x \sqrt {a +\frac {b}{x^{4}}}}-\frac {5 \sqrt {\frac {\cos \left (4 \,\mathrm {arccot}\left (\frac {a^{\frac {1}{4}} x}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \,\mathrm {arccot}\left (\frac {a^{\frac {1}{4}} x}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+\frac {\sqrt {b}}{x^{2}}\right ) \sqrt {\frac {a +\frac {b}{x^{4}}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^{2}}\right )^{2}}}}{24 \cos \left (2 \,\mathrm {arccot}\left (\frac {a^{\frac {1}{4}} x}{b^{\frac {1}{4}}}\right )\right ) a^{\frac {9}{4}} b^{\frac {1}{4}} \sqrt {a +\frac {b}{x^{4}}}} \]
command
integrate(1/(a+b/x^4)^(5/2)/x^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} {\rm ellipticF}\left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}, -1\right ) + {\left (7 \, a^{2} x^{7} + 5 \, a b x^{3}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, {\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {x^{10} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a^{3} x^{12} + 3 \, a^{2} b x^{8} + 3 \, a b^{2} x^{4} + b^{3}}, x\right ) \]