\[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^8} \, dx \]
Optimal antiderivative \[ \frac {\left (48 c^{3} d \,g^{3}-8 c^{2} g \left (3 b g \left (3 d h +e g \right )+a \left (3 d \,h^{2}-8 e g h +f \,g^{2}\right )\right )-b h \left (24 a^{2} f \,h^{2}-2 a b h \left (7 e h +10 f g \right )+b^{2} \left (9 d \,h^{2}+5 e g h +5 f \,g^{2}\right )\right )+2 c \left (4 a^{2} h^{2} \left (-e h +8 f g \right )-2 a b h \left (-3 d \,h^{2}+13 e g h +13 f \,g^{2}\right )+b^{2} g \left (21 d \,h^{2}+10 e g h +7 f \,g^{2}\right )\right )\right ) \left (b g -2 a h +\left (-b h +2 c g \right ) x \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{384 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{4} \left (h x +g \right )^{4}}-\frac {\left (f \,g^{2}-h \left (-d h +e g \right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{7 h \left (a \,h^{2}-b g h +c \,g^{2}\right ) \left (h x +g \right )^{7}}+\frac {\left (2 c g \left (5 f \,g^{2}+h \left (-9 d h +2 e g \right )\right )+h \left (14 a h \left (-e h +2 f g \right )-b \left (-9 d \,h^{2}-5 e g h +19 f \,g^{2}\right )\right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{84 h \left (a \,h^{2}-b g h +c \,g^{2}\right )^{2} \left (h x +g \right )^{6}}+\frac {\left (4 c^{2} g^{2} \left (5 f \,g^{2}+h \left (-51 d h +2 e g \right )\right )-7 h^{2} \left (24 a^{2} f \,h^{2}-2 a b h \left (7 e h +10 f g \right )+b^{2} \left (9 d \,h^{2}+5 e g h +5 f \,g^{2}\right )\right )-2 c h \left (3 b g \left (-34 d \,h^{2}-15 e g h +8 f \,g^{2}\right )-2 a h \left (12 d \,h^{2}-61 e g h +26 f \,g^{2}\right )\right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{840 h \left (a \,h^{2}-b g h +c \,g^{2}\right )^{3} \left (h x +g \right )^{5}}+\frac {\left (-4 a c +b^{2}\right )^{2} \left (48 c^{3} d \,g^{3}-8 c^{2} g \left (3 b g \left (3 d h +e g \right )+a \left (3 d \,h^{2}-8 e g h +f \,g^{2}\right )\right )-b h \left (24 a^{2} f \,h^{2}-2 a b h \left (7 e h +10 f g \right )+b^{2} \left (9 d \,h^{2}+5 e g h +5 f \,g^{2}\right )\right )+2 c \left (4 a^{2} h^{2} \left (-e h +8 f g \right )-2 a b h \left (-3 d \,h^{2}+13 e g h +13 f \,g^{2}\right )+b^{2} g \left (21 d \,h^{2}+10 e g h +7 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 )}{2048 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{\frac {11}{2}}}-\frac {\left (-4 a c +b^{2}\right ) \left (48 c^{3} d \,g^{3}-8 c^{2} g \left (3 b g \left (3 d h +e g \right )+a \left (3 d \,h^{2}-8 e g h +f \,g^{2}\right )\right )-b h \left (24 a^{2} f \,h^{2}-2 a b h \left (7 e h +10 f g \right )+b^{2} \left (9 d \,h^{2}+5 e g h +5 f \,g^{2}\right )\right )+2 c \left (4 a^{2} h^{2} \left (-e h +8 f g \right )-2 a b h \left (-3 d \,h^{2}+13 e g h +13 f \,g^{2}\right )+b^{2} g \left (21 d \,h^{2}+10 e g h +7 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}}{1024 \left (a \,h^{2}-b g h +c \,g^{2}\right )^{5} \left (h x +g \right )^{2}} \]
command
integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^8,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 \[ \text {Timed out} \]_______________________________________________________