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