\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {e^{4} x}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e \,x^{2}+d \right )}+\frac {x \left (a b c e \left (-b e +2 c d \right )+\left (-2 a c +b^{2}\right ) \left (c^{2} d^{2}+b^{2} e^{2}-c e \left (a e +2 b d \right )\right )-c \left (2 b^{2} c d e -4 a \,c^{2} d e -b^{3} e^{2}-b c \left (-3 a \,e^{2}+c \,d^{2}\right )\right ) x^{2}\right )}{2 a \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {e^{\frac {7}{2}} \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {2 e^{\frac {7}{2}} \left (-b e +2 c d \right ) \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {d}}+\frac {e^{2} \arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \sqrt {c}\, \left (3 c^{2} d^{2}+b \,e^{2} \left (b +\sqrt {-4 a c +b^{2}}\right )-c e \left (3 b d +a e +2 d \sqrt {-4 a c +b^{2}}\right )\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {-4 a c +b^{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^{4} e^{2}-b^{3} e \left (2 c d -e \sqrt {-4 a c +b^{2}}\right )-4 a \,c^{2} \left (3 c \,d^{2}-e \left (3 a e +d \sqrt {-4 a c +b^{2}}\right )\right )-b c \left (3 a \,e^{2} \sqrt {-4 a c +b^{2}}-c d \left (16 a e +d \sqrt {-4 a c +b^{2}}\right )\right )+b^{2} c \left (c \,d^{2}-e \left (9 a e +2 d \sqrt {-4 a c +b^{2}}\right )\right )\right ) \sqrt {2}}{4 a \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \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 {2}\, \sqrt {c}\, \left (3 c^{2} d^{2}+b \,e^{2} \left (b -\sqrt {-4 a c +b^{2}}\right )-c e \left (3 b d +a e -2 d \sqrt {-4 a c +b^{2}}\right )\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {-4 a c +b^{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^{4} e^{2}-b^{3} e \left (2 c d +e \sqrt {-4 a c +b^{2}}\right )+b c \left (3 a \,e^{2} \sqrt {-4 a c +b^{2}}-c d \left (-16 a e +d \sqrt {-4 a c +b^{2}}\right )\right )-4 a \,c^{2} \left (3 c \,d^{2}+e \left (-3 a e +d \sqrt {-4 a c +b^{2}}\right )\right )+b^{2} c \left (c \,d^{2}+e \left (-9 a e +2 d \sqrt {-4 a c +b^{2}}\right )\right )\right ) \sqrt {2}}{4 a \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(1/(e*x^2+d)^2/(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 \[ \text {Timed out} \]_______________________________________________________