\[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (2 A \left (a^{2} b^{2} f^{2}+4 b^{4} e^{2}\right )+a^{2} \left (a^{2} C \,f^{2}+2 b^{2} e \left (2 B f +C e \right )\right )\right ) x \sqrt {b x +a}\, \sqrt {-b c x +a c}}{16 b^{4}}+\frac {\left (-2 B f +C e \right ) \left (f x +e \right )^{2} \left (-b^{2} x^{2}+a^{2}\right ) \sqrt {b x +a}\, \sqrt {-b c x +a c}}{10 b^{2} f}-\frac {C \left (f x +e \right )^{3} \left (-b^{2} x^{2}+a^{2}\right ) \sqrt {b x +a}\, \sqrt {-b c x +a c}}{6 b^{2} f}-\frac {\left (16 a^{2} f^{2} \left (B f +2 C e \right )-8 b^{2} e \left (C \,e^{2}-2 f \left (5 A f +B e \right )\right )+3 f \left (5 a^{2} C \,f^{2}-b^{2} \left (2 C \,e^{2}-2 f \left (5 A f +2 B e \right )\right )\right ) x \right ) \left (-b^{2} x^{2}+a^{2}\right ) \sqrt {b x +a}\, \sqrt {-b c x +a c}}{120 b^{4} f}+\frac {a^{2} \left (2 A \left (a^{2} b^{2} f^{2}+4 b^{4} e^{2}\right )+a^{2} \left (a^{2} C \,f^{2}+2 b^{2} e \left (2 B f +C e \right )\right )\right ) \arctan \left (\frac {b x \sqrt {c}}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b c x +a c}}{16 b^{5} \sqrt {-b^{2} c \,x^{2}+a^{2} c}} \]
command
integrate((f*x+e)^2*(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 \[ \text {Timed out} \]_______________________________________________________