\[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-2 b g +c f \right ) x}{c^{3}}+\frac {g \,x^{3}}{3 c^{2}}+\frac {x \left (a \left (2 c^{3} d -c^{2} \left (2 a f +b e \right )-b^{3} g +b c \left (3 a g +b f \right )\right )+\left (b^{3} c f +b \,c^{2} \left (-3 a f +c d \right )-b^{4} g -b^{2} c \left (-4 a g +c e \right )+2 a \,c^{2} \left (-a g +c e \right )\right ) x^{2}\right )}{2 c^{3} \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (3 b^{3} c f -b \,c^{2} \left (13 a f +c d \right )-5 b^{4} g -b^{2} c \left (-24 a g +c e \right )+2 a \,c^{2} \left (-7 a g +3 c e \right )+\frac {-3 b^{4} c f +4 a \,c^{3} \left (-5 a f +c d \right )+b^{2} c^{2} \left (19 a f +c d \right )+5 b^{5} g +b^{3} c \left (-34 a g +c e \right )-4 a b \,c^{2} \left (-13 a g +2 c e \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 c^{\frac {7}{2}} \left (-4 a c +b^{2}\right ) \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (3 b^{3} c f -b \,c^{2} \left (13 a f +c d \right )-5 b^{4} g -b^{2} c \left (-24 a g +c e \right )+2 a \,c^{2} \left (-7 a g +3 c e \right )+\frac {3 b^{4} c f -4 a \,c^{3} \left (-5 a f +c d \right )-b^{2} c^{2} \left (19 a f +c d \right )-5 b^{5} g -b^{3} c \left (-34 a g +c e \right )+4 a b \,c^{2} \left (-13 a g +2 c e \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 c^{\frac {7}{2}} \left (-4 a c +b^{2}\right ) \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(x^4*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,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 \[ \text {Timed out} \]