\[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {-b c \left (a j +c e \right )+a \,b^{2} l +2 a c \left (-a l +c g \right )-\left (2 c^{3} e -c^{2} \left (2 a j +b g \right )-b^{3} l +b c \left (3 a l +b j \right )\right ) x^{2}}{4 c^{2} \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {x \left (a b c \left (a k +c f \right )-b^{2} \left (a^{2} m +c^{2} d \right )+2 a c \left (a^{2} m -a c h +c^{2} d \right )+\left (a \,b^{2} c k +2 a \,c^{2} \left (-a k +c f \right )-a \,b^{3} m -b c \left (-3 a^{2} m +a c h +c^{2} d \right )\right ) x^{2}\right )}{4 a \,c^{2} \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {b^{3} j}{c}+2 b \left (a j +3 c e \right )-16 a^{2} l -\frac {b^{4} l}{c^{2}}-b^{2} \left (3 g -\frac {5 a l}{c}\right )+2 \left (-3 a b l +2 a c j +b^{2} j -3 b c g +6 c^{2} e \right ) x^{2}}{4 \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {x \left (4 a^{2} b \,c^{2} \left (a k +2 c f \right )+a \,b^{3} c \left (2 a k +c f \right )-a \,b^{2} c \left (-11 a^{2} m +7 a c h +25 c^{2} d \right )+4 a^{2} c^{2} \left (-9 a^{2} m +a c h +7 c^{2} d \right )+b^{4} \left (-2 a^{2} m +3 c^{2} d \right )+c \left (a \,b^{2} c \left (3 a k +c f \right )+4 a^{2} c^{2} \left (3 a k +5 c f \right )+b^{3} \left (a^{2} m +3 c^{2} d \right )-4 a b c \left (4 a^{2} m +3 a c h +6 c^{2} d \right )\right ) x^{2}\right )}{8 a^{2} c^{2} \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\left (-3 a b l +2 a c j +b^{2} j -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 ) \left (a \,b^{2} c \left (3 a k +c f \right )+4 a^{2} c^{2} \left (3 a k +5 c f \right )+b^{3} \left (a^{2} m +3 c^{2} d \right )-4 a b c \left (4 a^{2} m +3 a c h +6 c^{2} d \right )+\frac {a \,b^{3} c \left (-3 a k +c f \right )-4 a^{2} b \,c^{2} \left (9 a k +13 c f \right )-6 a \,b^{2} c \left (-3 a^{2} m -3 a c h +5 c^{2} d \right )+b^{4} \left (-a^{2} m +3 c^{2} d \right )+8 a^{2} c^{2} \left (5 a^{2} m +3 a c h +21 c^{2} d \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{16 a^{2} c^{\frac {3}{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 ) \left (a \,b^{2} c \left (3 a k +c f \right )+4 a^{2} c^{2} \left (3 a k +5 c f \right )+b^{3} \left (a^{2} m +3 c^{2} d \right )-4 a b c \left (4 a^{2} m +3 a c h +6 c^{2} d \right )+\frac {-a \,b^{3} c \left (-3 a k +c f \right )+4 a^{2} b \,c^{2} \left (9 a k +13 c f \right )+6 a \,b^{2} c \left (-3 a^{2} m -3 a c h +5 c^{2} d \right )-b^{4} \left (-a^{2} m +3 c^{2} d \right )-8 a^{2} c^{2} \left (5 a^{2} m +3 a c h +21 c^{2} d \right )}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{16 a^{2} c^{\frac {3}{2}} \left (-4 a c +b^{2}\right )^{2} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate((m*x^8+l*x^7+k*x^6+j*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 \[ \text {Timed out} \]_______________________________________________________