\[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx \]
Optimal antiderivative \[ -\frac {5 a x \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b}+\frac {5 a^{\frac {7}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {b}\right ) \sqrt {\frac {b \,x^{4}+a}{\left (\sqrt {a}+x^{2} \sqrt {b}\right )^{2}}}}{42 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} \sqrt {b \,x^{4}+a}} \]
command
integrate(x^8/(b*x^4+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, a \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}, -1\right ) + {\left (3 \, b x^{5} - 5 \, a x\right )} \sqrt {b x^{4} + a}}{21 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {x^{8}}{\sqrt {b x^{4} + a}}, x\right ) \]