\[ \int \frac {x^8}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {3 d^{2} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{8 c \,e^{\frac {5}{2}}}+\frac {b d \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{2 c^{2} e^{\frac {3}{2}}}+\frac {\left (-a c +b^{2}\right ) \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{c^{3} \sqrt {e}}-\frac {3 d x \sqrt {e \,x^{2}+d}}{8 c \,e^{2}}-\frac {b x \sqrt {e \,x^{2}+d}}{2 c^{2} e}+\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 c e}-\frac {\arctan \left (\frac {x \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}{\sqrt {e \,x^{2}+d}\, \sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{3}-2 a b c +\frac {-2 a^{2} c^{2}+4 a \,b^{2} c -b^{4}}{\sqrt {-4 a c +b^{2}}}\right )}{c^{3} \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}\, \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {\arctan \left (\frac {x \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}{\sqrt {e \,x^{2}+d}\, \sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{3}-2 a b c +\frac {2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}}{\sqrt {-4 a c +b^{2}}}\right )}{c^{3} \sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate(x^8/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]