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