\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-c f +d e \right ) \left (b^{2} \left (A \,d^{2} e^{2}-2 c d e \left (-A f +B e \right )+c^{2} \left (5 A \,f^{2}-6 B e f +8 C \,e^{2}\right )\right )+a b \left (d^{2} e \left (-4 A f +B e \right )-c^{2} f \left (-B f +4 C e \right )-2 c d \left (6 A \,f^{2}-7 B e f +6 C \,e^{2}\right )\right )-a^{2} \left (2 d f \left (-4 A d f +B c f +3 B d e \right )-C \left (c^{2} f^{2}+2 c d e f +5 d^{2} e^{2}\right )\right )\right ) \arctanh \left (\frac {\sqrt {-a f +b e}\, \sqrt {d x +c}}{\sqrt {-a d +b c}\, \sqrt {f x +e}}\right )}{8 \left (-a d +b c \right )^{\frac {5}{2}} \left (-a f +b e \right )^{\frac {7}{2}}}-\frac {\left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {f x +e}}{3 b \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b x +a \right )^{3}}+\frac {\left (4 a^{3} C d f -b^{3} \left (-5 A c f -3 A d e +6 B c e \right )+a \,b^{2} \left (-8 A d f +B c f +3 B d e +12 c C e \right )-a^{2} b \left (-2 B d f +7 c C f +9 C d e \right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{12 b^{2} \left (-a d +b c \right ) \left (-a f +b e \right )^{2} \left (b x +a \right )^{2}}-\frac {\left (8 a^{4} C \,d^{2} f^{2}-2 a^{3} b d f \left (-2 B d f +7 c C f +13 C d e \right )-b^{4} \left (3 A \,d^{2} e^{2}-2 c d e \left (-2 A f +3 B e \right )-3 c^{2} \left (5 A \,f^{2}-6 B e f +8 C \,e^{2}\right )\right )-a \,b^{3} \left (d^{2} e \left (-10 A f +3 B e \right )+3 c^{2} f \left (-B f +4 C e \right )+2 c d \left (13 A \,f^{2}-14 B e f +30 C \,e^{2}\right )\right )-a^{2} b^{2} \left (4 d f \left (-2 A d f +B c f +4 B d e \right )-C \left (3 c^{2} f^{2}+44 c d e f +33 d^{2} e^{2}\right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{24 b^{2} \left (-a d +b c \right )^{2} \left (-a f +b e \right )^{3} \left (b x +a \right )} \]
command
integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^4/(f*x+e)^(1/2),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 \[ \text {Timed out} \]_______________________________________________________