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