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