\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{\sqrt {a+b x}} \, dx \]
Optimal antiderivative \[ \frac {2 C \left (d x +c \right )^{\frac {3}{2}} \left (f x +e \right )^{\frac {3}{2}} \sqrt {b x +a}}{7 b d f}-\frac {2 \left (6 a C d f -b \left (7 B d f -4 C \left (c f +d e \right )\right )\right ) \left (f x +e \right )^{\frac {3}{2}} \sqrt {b x +a}\, \sqrt {d x +c}}{35 b^{2} d \,f^{2}}-\frac {2 \left (5 b d f \left (3 a C \left (c f +d e \right )+b \left (-7 A d f +c C e \right )\right )-\left (4 a d f -b c f +2 b d e \right ) \left (6 a C d f -b \left (7 B d f -4 C \left (c f +d e \right )\right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{105 b^{3} d^{2} f^{2}}-\frac {2 \left (3 b d f \left (5 b c f \left (3 a C \left (c f +d e \right )+b \left (-7 A d f +c C e \right )\right )-\left (3 a c f +a d e +b c e \right ) \left (6 a C d f -b \left (7 B d f -4 C \left (c f +d e \right )\right )\right )\right )+2 \left (\frac {b d e}{2}-\left (a d +b c \right ) f \right ) \left (5 b d f \left (3 a C \left (c f +d e \right )+b \left (-7 A d f +c C e \right )\right )-\left (4 a d f -b c f +2 b d e \right ) \left (6 a C d f -b \left (7 B d f -4 C \left (c f +d e \right )\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}}{105 b^{4} d^{\frac {5}{2}} f^{3} \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 (24 a^{2} C \,d^{2} f^{2}+a b d f \left (-28 B d f -5 c C f +13 C d e \right )-b^{2} \left (7 d f \left (-5 A d f -B c f +2 B d e \right )-C \left (-4 c^{2} f^{2}-c d e f +8 d^{2} e^{2}\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}}}{105 b^{4} d^{\frac {5}{2}} f^{3} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, {\left (15 \, C b^{4} d^{4} f^{4} x^{2} - 4 \, C b^{4} d^{4} f^{2} e^{2} + 3 \, {\left (C b^{4} c d^{3} - {\left (6 \, C a b^{3} - 7 \, B b^{4}\right )} d^{4}\right )} f^{4} x - {\left (4 \, C b^{4} c^{2} d^{2} + {\left (5 \, C a b^{3} - 7 \, B b^{4}\right )} c d^{3} - {\left (24 \, C a^{2} b^{2} - 28 \, B a b^{3} + 35 \, A b^{4}\right )} d^{4}\right )} f^{4} + {\left (3 \, C b^{4} d^{4} f^{3} x + {\left (2 \, C b^{4} c d^{3} - {\left (5 \, C a b^{3} - 7 \, B b^{4}\right )} d^{4}\right )} f^{3}\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} - {\left (8 \, C b^{4} d^{4} e^{4} + {\left (8 \, C b^{4} c^{4} + {\left (5 \, C a b^{3} - 14 \, B b^{4}\right )} c^{3} d + {\left (10 \, C a^{2} b^{2} - 14 \, B a b^{3} + 35 \, A b^{4}\right )} c^{2} d^{2} + {\left (40 \, C a^{3} b - 49 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} c d^{3} - 2 \, {\left (24 \, C a^{4} - 28 \, B a^{3} b + 35 \, A a^{2} b^{2}\right )} d^{4}\right )} f^{4} - {\left (9 \, C b^{4} c^{3} d + 7 \, {\left (C a b^{3} - 3 \, B b^{4}\right )} c^{2} d^{2} + 14 \, {\left (3 \, C a^{2} b^{2} - 4 \, B a b^{3} + 10 \, A b^{4}\right )} c d^{3} - {\left (40 \, C a^{3} b - 49 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} d^{4}\right )} f^{3} e - {\left (4 \, C b^{4} c^{2} d^{2} + 7 \, {\left (C a b^{3} - 3 \, B b^{4}\right )} c d^{3} - {\left (10 \, C a^{2} b^{2} - 14 \, B a b^{3} + 35 \, A b^{4}\right )} d^{4}\right )} f^{2} e^{2} - {\left (9 \, C b^{4} c d^{3} - {\left (5 \, C a b^{3} - 14 \, B b^{4}\right )} d^{4}\right )} f e^{3}\right )} \sqrt {b d f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right ) - 3 \, {\left (8 \, C b^{4} d^{4} f e^{3} + {\left (8 \, C b^{4} c^{3} d + {\left (9 \, C a b^{3} - 14 \, B b^{4}\right )} c^{2} d^{2} + {\left (16 \, C a^{2} b^{2} - 21 \, B a b^{3} + 35 \, A b^{4}\right )} c d^{3} - 2 \, {\left (24 \, C a^{3} b - 28 \, B a^{2} b^{2} + 35 \, A a b^{3}\right )} d^{4}\right )} f^{4} - {\left (5 \, C b^{4} c^{2} d^{2} + 2 \, {\left (4 \, C a b^{3} - 7 \, B b^{4}\right )} c d^{3} - {\left (16 \, C a^{2} b^{2} - 21 \, B a b^{3} + 35 \, A b^{4}\right )} d^{4}\right )} f^{3} e - {\left (5 \, C b^{4} c d^{3} - {\left (9 \, C a b^{3} - 14 \, B b^{4}\right )} d^{4}\right )} f^{2} e^{2}\right )} \sqrt {b d f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2} - {\left (b^{2} c d + a b d^{2}\right )} f e\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3} - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{2} e - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f e^{2}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right )\right )}}{315 \, b^{5} d^{4} f^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{\sqrt {b x + a}}, x\right ) \]