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