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