\[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {e^{2} \left (6 c^{2} d^{2}+b^{2} e^{2}-c e \left (a e +4 b d \right )\right ) x}{c^{3}}+\frac {e^{3} \left (-b e +4 c d \right ) x^{3}}{3 c^{2}}+\frac {e^{4} x^{5}}{5 c}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (e \left (-b e +2 c d \right ) \left (2 c^{2} d^{2}+b^{2} e^{2}-2 c e \left (a e +b d \right )\right )+\frac {2 c^{4} d^{4}+b^{4} e^{4}-4 b^{2} c \,e^{3} \left (a e +b d \right )-4 c^{3} d^{2} e \left (3 a e +b d \right )+2 c^{2} e^{2} \left (a^{2} e^{2}+6 a b d e +3 b^{2} d^{2}\right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 c^{\frac {7}{2}} \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (e \left (-b e +2 c d \right ) \left (2 c^{2} d^{2}+b^{2} e^{2}-2 c e \left (a e +b d \right )\right )+\frac {-2 c^{4} d^{4}-b^{4} e^{4}+4 b^{2} c \,e^{3} \left (a e +b d \right )+4 c^{3} d^{2} e \left (3 a e +b d \right )-2 c^{2} e^{2} \left (a^{2} e^{2}+6 a b d e +3 b^{2} d^{2}\right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 c^{\frac {7}{2}} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate((e*x^2+d)^4/(c*x^4+b*x^2+a),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 \[ \text {Timed out} \]