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