\[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {x \left (c^{2} \left (a b f -b^{2} \left (d +\frac {a^{2} j}{c^{2}}\right )+2 a \left (c d -a h +\frac {a^{2} j}{c}\right )\right )+\left (2 a \,c^{3} f -a \,b^{3} j -b c \left (-3 a^{2} j +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 {-b \,c^{3} \left (a i +c e \right )+a \,b^{4} k -4 a^{2} b^{2} c k +2 a \,c^{2} \left (a^{2} k +c^{2} g \right )-\left (2 c^{5} e +b^{2} c^{3} i -c^{4} \left (2 a i +b g \right )-b^{5} k +5 a \,b^{3} c k -5 a^{2} b \,c^{2} k \right ) x^{2}}{4 c^{4} \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {x \left (c \left (a \,b^{3} f +8 a^{2} b c f +4 a^{2} \left (-9 a^{2} j +a c h +7 c^{2} d \right )+b^{4} \left (3 d -\frac {2 a^{2} j}{c^{2}}\right )-a \,b^{2} \left (25 c d +7 a h -\frac {11 a^{2} j}{c}\right )\right )+\left (a \,b^{2} c^{2} f +20 a^{2} c^{3} f +b^{3} \left (a^{2} j +3 c^{2} d \right )-4 a b c \left (4 a^{2} j +3 a c h +6 c^{2} d \right )\right ) x^{2}\right )}{8 a^{2} c \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {b^{3} c^{2} i +2 b \,c^{3} \left (a i +3 c e \right )+11 a \,b^{4} k -\frac {b^{6} k}{c}+32 a^{3} c^{2} k -3 b^{2} \left (13 a^{2} c k +c^{3} g \right )+2 \left (6 c^{5} e +b^{2} c^{3} i -c^{4} \left (-2 a i +3 b g \right )+2 b^{5} k -15 a \,b^{3} c k +25 a^{2} b \,c^{2} k \right ) x^{2}}{4 c^{3} \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\left (12 c^{5} e +2 b^{2} c^{3} i -c^{4} \left (-4 a i +6 b g \right )-b^{5} k +10 a \,b^{3} c k -30 a^{2} b \,c^{2} k \right ) \arctanh \left (\frac {2 c \,x^{2}+b}{\sqrt {-4 a c +b^{2}}}\right )}{2 c^{3} \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {k \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{3}}+\frac {\arctan \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \left (a \,b^{2} c^{2} f +20 a^{2} c^{3} f +b^{3} \left (a^{2} j +3 c^{2} d \right )-4 a b c \left (4 a^{2} j +3 a c h +6 c^{2} d \right )+\frac {a \,b^{3} c^{2} f -52 a^{2} b \,c^{3} f -6 a \,b^{2} c \left (-3 a^{2} j -3 a c h +5 c^{2} d \right )+b^{4} \left (-a^{2} j +3 c^{2} d \right )+8 a^{2} c^{2} \left (5 a^{2} j +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^{2} f +20 a^{2} c^{3} f +b^{3} \left (a^{2} j +3 c^{2} d \right )-4 a b c \left (4 a^{2} j +3 a c h +6 c^{2} d \right )+\frac {-a \,b^{3} c^{2} f +52 a^{2} b \,c^{3} f +6 a \,b^{2} c \left (-3 a^{2} j -3 a c h +5 c^{2} d \right )-b^{4} \left (-a^{2} j +3 c^{2} d \right )-8 a^{2} c^{2} \left (5 a^{2} j +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((k*x^11+j*x^8+i*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 \[ \text {Timed out} \]_______________________________________________________