\[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (f \,g^{2}-h \left (-d h +e g \right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 h \left (a \,h^{2}-b g h +c \,g^{2}\right ) \left (h x +g \right )^{5}}+\frac {\left (2 c g \left (3 f \,g^{2}+h \left (-7 d h +2 e g \right )\right )+h \left (10 a h \left (-e h +2 f g \right )-b \left (-7 d \,h^{2}-3 e g h +13 f \,g^{2}\right )\right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{40 h \left (a \,h^{2}-b g h +c \,g^{2}\right )^{2} \left (h x +g \right )^{4}}+\frac {\left (4 c^{2} g^{2} \left (3 f \,g^{2}+h \left (-27 d h +2 e g \right )\right )-5 h^{2} \left (16 a^{2} f \,h^{2}-2 a b h \left (5 e h +6 f g \right )+b^{2} \left (7 d \,h^{2}+3 e g h +3 f \,g^{2}\right )\right )-2 c h \left (b g \left (-54 d \,h^{2}-21 e g h +16 f \,g^{2}\right )-2 a h \left (8 d \,h^{2}-33 e g h +18 f \,g^{2}\right )\right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{240 h \left (a \,h^{2}-b g h +c \,g^{2}\right )^{3} \left (h x +g \right )^{3}}-\frac {\left (-4 a c +b^{2}\right ) \left (32 c^{3} d \,g^{3}-8 c^{2} g \left (2 b g \left (3 d h +e g \right )+a \left (3 d \,h^{2}-6 e g h +f \,g^{2}\right )\right )-b h \left (16 a^{2} f \,h^{2}-2 a b h \left (5 e h +6 f g \right )+b^{2} \left (7 d \,h^{2}+3 e g h +3 f \,g^{2}\right )\right )+2 c \left (4 a^{2} h^{2} \left (-e h +6 f g \right )-6 a b h \left (-d \,h^{2}+3 e g h +3 f \,g^{2}\right )+b^{2} g \left (15 d \,h^{2}+6 e g h +5 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 )}{256 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{\frac {9}{2}}}+\frac {\left (32 c^{3} d \,g^{3}-8 c^{2} g \left (2 b g \left (3 d h +e g \right )+a \left (3 d \,h^{2}-6 e g h +f \,g^{2}\right )\right )-b h \left (16 a^{2} f \,h^{2}-2 a b h \left (5 e h +6 f g \right )+b^{2} \left (7 d \,h^{2}+3 e g h +3 f \,g^{2}\right )\right )+2 c \left (4 a^{2} h^{2} \left (-e h +6 f g \right )-6 a b h \left (-d \,h^{2}+3 e g h +3 f \,g^{2}\right )+b^{2} g \left (15 d \,h^{2}+6 e g h +5 f \,g^{2}\right )\right )\right ) \left (b g -2 a h +\left (-b h +2 c g \right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{128 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{4} \left (h x +g \right )^{2}} \]
command
integrate((f*x^2+e*x+d)*(c*x^2+b*x+a)^(1/2)/(h*x+g)^6,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 \[ \text {Timed out} \]_______________________________________________________