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