\[ \int \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx \]
Optimal antiderivative \[ \frac {2 C \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (f x +e \right )^{\frac {3}{2}}}{9 b d f}-\frac {2 \left (2 a C d f -b \left (3 B d f -2 C \left (c f +d e \right )\right )\right ) \left (d x +c \right )^{\frac {3}{2}} \left (f x +e \right )^{\frac {3}{2}} \sqrt {b x +a}}{21 b \,d^{2} f^{2}}-\frac {2 \left (7 b d f \left (-3 A b d f +a c C f +a C d e +b c C e \right )+\left (a d f -4 b \left (c f +d e \right )\right ) \left (2 a C d f -b \left (3 B d f -2 C \left (c f +d e \right )\right )\right )\right ) \left (f x +e \right )^{\frac {3}{2}} \sqrt {b x +a}\, \sqrt {d x +c}}{105 b^{2} d^{2} f^{3}}+\frac {2 \left (8 a^{3} C \,d^{3} f^{3}+3 a^{2} b \,d^{2} f^{2} \left (-4 B d f -c C f +C d e \right )-3 a \,b^{2} d \,f^{2} \left (\left (-7 A \,d^{2}+c^{2} C \right ) f +B d \left (-2 c f +d e \right )\right )-b^{3} \left (C \left (-8 c^{3} f^{3}-3 c^{2} d e \,f^{2}+16 d^{3} e^{3}\right )+3 d f \left (7 A d f \left (-c f +2 d e \right )-B \left (-4 c^{2} f^{2}-c d e f +8 d^{2} e^{2}\right )\right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{315 b^{3} d^{3} f^{3}}-\frac {2 \left (16 a^{4} C \,d^{4} f^{4}-8 a^{3} b \,d^{3} f^{3} \left (3 B d f +c C f +C d e \right )+3 a^{2} b^{2} d^{2} f^{2} \left (d f \left (14 A d f +5 B c f +5 B d e \right )-2 C \left (c^{2} f^{2}-c d e f +d^{2} e^{2}\right )\right )-a \,b^{3} d f \left (C \left (8 c^{3} f^{3}-6 c^{2} d e \,f^{2}-6 c \,d^{2} e^{2} f +8 d^{3} e^{3}\right )+3 d f \left (14 A d f \left (c f +d e \right )-B \left (5 c^{2} f^{2}-6 c d e f +5 d^{2} e^{2}\right )\right )\right )+b^{4} \left (2 C \left (8 c^{4} f^{4}-4 c^{3} d e \,f^{3}-3 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +8 d^{4} e^{4}\right )+3 d f \left (14 A d f \left (c^{2} f^{2}-c d e f +d^{2} e^{2}\right )-B \left (8 c^{3} f^{3}-5 c^{2} d e \,f^{2}-5 c \,d^{2} e^{2} f +8 d^{3} e^{3}\right )\right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{315 b^{4} d^{\frac {7}{2}} f^{4} \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}-\frac {2 \left (-a f +b e \right ) \left (-c f +d e \right ) \left (8 a^{3} C \,d^{3} f^{3}+3 a^{2} b \,d^{2} f^{2} \left (-4 B d f -c C f +C d e \right )-3 a \,b^{2} d \,f^{2} \left (\left (-7 A \,d^{2}+c^{2} C \right ) f +B d \left (-2 c f +d e \right )\right )-b^{3} \left (C \left (-8 c^{3} f^{3}-3 c^{2} d e \,f^{2}+16 d^{3} e^{3}\right )+3 d f \left (7 A d f \left (-c f +2 d e \right )-B \left (-4 c^{2} f^{2}-c d e f +8 d^{2} e^{2}\right )\right )\right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{315 b^{4} d^{\frac {7}{2}} f^{4} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((b*x+a)^(1/2)*(C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left ({\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}, x\right ) \]