\[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (4 A \,b^{2} e +a^{2} \left (B f +C e \right )\right ) x \sqrt {b x +a}\, \sqrt {-b c x +a c}}{8 b^{2}}-\frac {C \left (f x +e \right )^{2} \left (-b^{2} x^{2}+a^{2}\right ) \sqrt {b x +a}\, \sqrt {-b c x +a c}}{5 b^{2} f}-\frac {\left (8 a^{2} C \,f^{2}-4 b^{2} \left (3 C \,e^{2}-5 f \left (A f +B e \right )\right )-3 b^{2} f \left (-5 B f +3 C e \right ) x \right ) \left (-b^{2} x^{2}+a^{2}\right ) \sqrt {b x +a}\, \sqrt {-b c x +a c}}{60 b^{4} f}+\frac {a^{2} \left (4 A \,b^{2} e +a^{2} \left (B f +C e \right )\right ) \arctan \left (\frac {b x \sqrt {c}}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b c x +a c}}{8 b^{3} \sqrt {-b^{2} c \,x^{2}+a^{2} c}} \]
command
integrate((f*x+e)*(C*x^2+B*x+A)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(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 \[ \text {Timed out} \]_______________________________________________________