\[ \int \frac {x^8}{\left (a+b x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {x^{5}}{2 b \sqrt {b \,x^{4}+a}}+\frac {5 x \sqrt {b \,x^{4}+a}}{6 b^{2}}-\frac {5 a^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {b}\right ) \sqrt {\frac {b \,x^{4}+a}{\left (\sqrt {a}+x^{2} \sqrt {b}\right )^{2}}}}{12 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} \sqrt {b \,x^{4}+a}} \]
command
integrate(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
\[ -\frac {5 \, {\left (b x^{4} + a\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}, -1\right ) - {\left (2 \, b x^{5} + 5 \, a x\right )} \sqrt {b x^{4} + a}}{6 \, {\left (b^{3} x^{4} + a b^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} x^{8}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]