\[ \int \frac {1}{x^8 \left (a+b x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{2 a \,x^{7} \sqrt {b \,x^{4}+a}}-\frac {9 \sqrt {b \,x^{4}+a}}{14 a^{2} x^{7}}+\frac {15 b \sqrt {b \,x^{4}+a}}{14 a^{3} x^{3}}+\frac {15 b^{\frac {7}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {b}\right ) \sqrt {\frac {b \,x^{4}+a}{\left (\sqrt {a}+x^{2} \sqrt {b}\right )^{2}}}}{28 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {13}{4}} \sqrt {b \,x^{4}+a}} \]
command
integrate(1/x^8/(b*x^4+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, {\left (b^{2} x^{11} + a b x^{7}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} {\rm ellipticF}\left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}, -1\right ) - {\left (15 \, b^{2} x^{8} + 6 \, a b x^{4} - 2 \, a^{2}\right )} \sqrt {b x^{4} + a}}{14 \, {\left (a^{3} b x^{11} + a^{4} x^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a}}{b^{2} x^{16} + 2 \, a b x^{12} + a^{2} x^{8}}, x\right ) \]