\[ \int \frac {1}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {x \left (b^{2} c d -2 a \,c^{2} d -b^{3} e +3 a b c e +c \left (2 a c e -b^{2} e +b c d \right ) x^{2}\right )}{2 a \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {e^{\frac {7}{2}} \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {d}}-\frac {e^{2} \arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (e +\frac {b e -2 c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (b c d -b^{2} e +2 a c e +\frac {8 a b c e -12 a \,c^{2} d -b^{3} e +b^{2} c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 a \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {e^{2} \arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (e +\frac {-b e +2 c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {b +\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (b c d -b^{2} e +2 a c e +\frac {-8 a b c e +12 a \,c^{2} d +b^{3} e -b^{2} c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 a \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(1/(e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________