\[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (a+b x^2+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x \left (b^{2} d -a b f -2 a \left (-a h +c d \right )+\left (a b h -2 a c f +b c d \right ) x^{2}\right )}{4 a \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {2 a c g -b \left (a i +c e \right )-\left (-2 a c i +b^{2} i -b c g +2 c^{2} e \right ) x^{2}}{4 c \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (6 c e -3 b g +2 a i +\frac {b^{2} i}{c}\right ) \left (2 c \,x^{2}+b \right )}{4 \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {x \left (3 b^{4} d +a \,b^{3} f +8 a^{2} b c f +4 a^{2} c \left (a h +7 c d \right )-a \,b^{2} \left (7 a h +25 c d \right )+c \left (3 b^{3} d +a \,b^{2} f +20 a^{2} c f -12 a b \left (a h +2 c d \right )\right ) x^{2}\right )}{8 a^{2} \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\left (2 a c i +b^{2} i -3 b c g +6 c^{2} e \right ) \arctanh \left (\frac {2 c \,x^{2}+b}{\sqrt {-4 a c +b^{2}}}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (3 b^{3} d +a \,b^{2} f +20 a^{2} c f -12 a b \left (a h +2 c d \right )+\frac {3 b^{4} d +a \,b^{3} f -52 a^{2} b c f -6 a \,b^{2} \left (-3 a h +5 c d \right )+24 a^{2} c \left (a h +7 c d \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{16 a^{2} \left (-4 a c +b^{2}\right )^{2} \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {c}\, \left (3 b^{3} d +a \,b^{2} f +20 a^{2} c f -12 a b \left (a h +2 c d \right )+\frac {-3 b^{4} d -a \,b^{3} f +52 a^{2} b c f +6 a \,b^{2} \left (-3 a h +5 c d \right )-24 a^{2} c \left (a h +7 c d \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{16 a^{2} \left (-4 a c +b^{2}\right )^{2} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,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 \[ \text {Timed out} \]_______________________________________________________