\[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (8 c^{2} g^{2} \left (6 d \,h^{2}-3 e g h +f \,g^{2}\right )+h^{2} \left (8 a^{2} f \,h^{2}+4 a b h \left (-3 e h +2 f g \right )-b^{2} \left (f \,g^{2}+3 h \left (-5 d h +e g \right )\right )\right )-4 c h \left (a h \left (3 d \,h^{2}-9 e g h +11 f \,g^{2}\right )-b g \left (2 f \,g^{2}+3 h \left (-4 d h +e g \right )\right )\right )\right ) \arctanh \left (\frac {b g -2 a h +\left (-b h +2 c g \right ) x}{2 \sqrt {a \,h^{2}-b g h +c \,g^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{8 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{\frac {7}{2}}}+\frac {2 b^{4} d \,h^{3}-2 b^{3} h^{2} \left (a e h +3 c d g \right )+2 b^{2} h \left (3 c^{2} d \,g^{2}+a^{2} f \,h^{2}+a c h \left (-4 d h +3 e g \right )\right )-2 b c \left (c^{2} d \,g^{3}+3 a^{2} h^{2} \left (-e h +f g \right )+a c g \left (-9 d \,h^{2}+3 e g h +f \,g^{2}\right )\right )-4 a c \left (a^{2} f \,h^{3}-c^{2} g^{2} \left (-3 d h +e g \right )-a c h \left (d \,h^{2}-3 e g h +3 f \,g^{2}\right )\right )-2 c \left (2 c^{3} d \,g^{3}-b \left (a^{2} f -a b e +b^{2} d \right ) h^{3}-c^{2} g \left (b g \left (3 d h +e g \right )+2 a \left (3 d \,h^{2}-3 e g h +f \,g^{2}\right )\right )+c \left (2 a^{2} h^{2} \left (-e h +3 f g \right )-3 a b h \left (-d \,h^{2}+e g h +f \,g^{2}\right )+b^{2} \left (3 d g \,h^{2}+f \,g^{3}\right )\right )\right ) x}{\left (-4 a c +b^{2}\right ) \left (a \,h^{2}-b g h +c \,g^{2}\right )^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {h \left (f \,g^{2}-h \left (-d h +e g \right )\right ) \sqrt {c \,x^{2}+b x +a}}{2 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{2} \left (h x +g \right )^{2}}-\frac {h \left (2 c g \left (3 f \,g^{2}-h \left (-7 d h +5 e g \right )\right )-h \left (4 a h \left (-e h +2 f g \right )-b \left (-7 d \,h^{2}+3 e g h +f \,g^{2}\right )\right )\right ) \sqrt {c \,x^{2}+b x +a}}{4 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{3} \left (h x +g \right )} \]
command
integrate((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+b*x+a)^(3/2),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 \[ \text {Timed out} \]