\[ \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ \frac {\left (b^{3} \left (5 A \,d^{3} e^{3}-3 c \,d^{2} e^{2} \left (-A f +2 B e \right )+c^{2} d e \left (3 A \,f^{2}-4 B e f +8 C \,e^{2}\right )+c^{3} f \left (5 A \,f^{2}-6 B e f +8 C \,e^{2}\right )\right )+a \,b^{2} \left (d^{3} e^{2} \left (-18 A f +B e \right )-c^{3} f^{2} \left (-B f +4 C e \right )-c \,d^{2} e \left (12 A \,f^{2}-23 B e f +4 C \,e^{2}\right )-c^{2} d f \left (18 A \,f^{2}-23 B e f +40 C \,e^{2}\right )\right )-2 a^{3} d f \left (C \left (3 c^{2} f^{2}+2 c d e f +3 d^{2} e^{2}\right )+4 d f \left (2 A d f -B \left (c f +d e \right )\right )\right )+a^{2} b \left (C \left (c^{3} f^{3}+23 c^{2} d e \,f^{2}+23 c \,d^{2} e^{2} f +d^{3} e^{3}\right )+4 d f \left (6 A d f \left (c f +d e \right )-B \left (c^{2} f^{2}+10 c d e f +d^{2} e^{2}\right )\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 {7}{2}} \left (-a f +b e \right )^{\frac {7}{2}}}-\frac {\left (A \,b^{2}-a \left (b B -a C \right )\right ) \sqrt {d x +c}\, \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 (2 a^{3} C d f +a \,b^{2} \left (-10 A d f +B c f +B d e +12 c C e \right )-b^{3} \left (6 B c e -5 A \left (c f +d e \right )\right )+a^{2} b \left (4 B d f -7 C \left (c f +d e \right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{12 b \left (-a d +b c \right )^{2} \left (-a f +b e \right )^{2} \left (b x +a \right )^{2}}+\frac {\left (4 a^{4} C \,d^{2} f^{2}+8 a^{3} b d f \left (B d f -2 C \left (c f +d e \right )\right )-b^{4} \left (15 A \,d^{2} e^{2}-2 c d e \left (-7 A f +9 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 (-44 A f +3 B e \right )-3 c^{2} f \left (-B f +4 C e \right )-2 c d \left (22 A \,f^{2}-29 B e f +6 C \,e^{2}\right )\right )-a^{2} b^{2} \left (C \left (3 c^{2} f^{2}-34 c d e f +3 d^{2} e^{2}\right )+2 d f \left (22 A d f -5 B \left (c f +d e \right )\right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{24 b \left (-a d +b c \right )^{3} \left (-a f +b e \right )^{3} \left (b x +a \right )} \]
command
integrate((C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2)/(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 \[ \text {Timed out} \]_______________________________________________________