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