\[ \int \frac {x^6 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-2 b f +c e \right ) x}{c^{3}}+\frac {f \,x^{3}}{3 c^{2}}+\frac {x \left (a \left (b^{2} c e -2 a \,c^{2} e -b^{3} f -b c \left (-3 a f +c d \right )\right )+\left (b^{3} c e -3 a b \,c^{2} e -b^{4} f -b^{2} c \left (-4 a f +c d \right )+2 a \,c^{2} \left (-a f +c d \right )\right ) x^{2}\right )}{2 c^{3} \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (3 b^{3} c e -13 a b \,c^{2} e -5 b^{4} f -b^{2} c \left (-24 a f +c d \right )+2 a \,c^{2} \left (-7 a f +3 c d \right )+\frac {-3 b^{4} c e +19 a \,b^{2} c^{2} e -20 a^{2} c^{3} e +5 b^{5} f +b^{3} c \left (-34 a f +c d \right )-4 a b \,c^{2} \left (-13 a f +2 c d \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 c^{\frac {7}{2}} \left (-4 a c +b^{2}\right ) \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \left (3 b^{3} c e -13 a b \,c^{2} e -5 b^{4} f -b^{2} c \left (-24 a f +c d \right )+2 a \,c^{2} \left (-7 a f +3 c d \right )+\frac {3 b^{4} c e -19 a \,b^{2} c^{2} e +20 a^{2} c^{3} e -5 b^{5} f -b^{3} c \left (-34 a f +c d \right )+4 a b \,c^{2} \left (-13 a f +2 c d \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{4 c^{\frac {7}{2}} \left (-4 a c +b^{2}\right ) \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(x^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 \[ \text {Timed out} \]