\[ \int \frac {1}{x^4 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {e \,x^{2}+d}}{3 a d \,x^{3}}+\frac {b \sqrt {e \,x^{2}+d}}{a^{2} d x}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 a \,d^{2} x}+\frac {c \arctan \left (\frac {x \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}{\sqrt {e \,x^{2}+d}\, \sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (b +\frac {-2 a c +b^{2}}{\sqrt {-4 a c +b^{2}}}\right )}{a^{2} \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}\, \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {c \arctan \left (\frac {x \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}{\sqrt {e \,x^{2}+d}\, \sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (b +\frac {2 a c -b^{2}}{\sqrt {-4 a c +b^{2}}}\right )}{a^{2} \sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate(1/x^4/(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 \[ \text {Timed out} \]