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