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