\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \left (f x +e \right )^{\frac {3}{2}}}{b \left (-a d +b c \right ) \left (-a f +b e \right ) \sqrt {b x +a}}+\frac {2 \left (6 a^{2} C d f +b^{2} \left (5 A d f +c C e \right )-a b \left (5 B d f +c C f +C d e \right )\right ) \left (f x +e \right )^{\frac {3}{2}} \sqrt {b x +a}\, \sqrt {d x +c}}{5 b^{2} \left (-a d +b c \right ) f \left (-a f +b e \right )}+\frac {2 \left (24 a^{2} C d \,f^{2}-a b f \left (20 B d f +c C f +7 C d e \right )+b^{2} \left (5 d f \left (3 A f +B e \right )-C e \left (-c f +2 d e \right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{15 b^{3} d f \left (-a f +b e \right )}+\frac {2 \left (48 a^{2} C \,d^{2} f^{2}-8 a b d f \left (5 B d f +c C f +C d e \right )+b^{2} \left (5 d f \left (6 A d f +B c f +B d e \right )-2 C \left (c^{2} f^{2}-c d e f +d^{2} e^{2}\right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{15 b^{4} d^{\frac {3}{2}} f^{2} \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}-\frac {2 \left (-c f +d e \right ) \left (24 a^{2} C d \,f^{2}-a b f \left (20 B d f +c C f +7 C d e \right )+b^{2} \left (5 d f \left (3 A f +B e \right )-C e \left (-c f +2 d e \right )\right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{15 b^{4} d^{\frac {3}{2}} f^{2} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, {\left (3 \, C b^{4} d^{3} f^{3} x^{2} + {\left (C b^{4} c d^{2} - {\left (6 \, C a b^{3} - 5 \, B b^{4}\right )} d^{3}\right )} f^{3} x + {\left (C a b^{3} c d^{2} - {\left (24 \, C a^{2} b^{2} - 20 \, B a b^{3} + 15 \, A b^{4}\right )} d^{3}\right )} f^{3} + {\left (C b^{4} d^{3} f^{2} x + C a b^{3} d^{3} f^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} + {\left ({\left (2 \, C b^{4} c^{3} + {\left (7 \, C a b^{3} - 5 \, B b^{4}\right )} c^{2} d + {\left (32 \, C a^{2} b^{2} - 25 \, B a b^{3} + 15 \, A b^{4}\right )} c d^{2} - 2 \, {\left (24 \, C a^{3} b - 20 \, B a^{2} b^{2} + 15 \, A a b^{3}\right )} d^{3}\right )} f^{3} x + {\left (2 \, C a b^{3} c^{3} + {\left (7 \, C a^{2} b^{2} - 5 \, B a b^{3}\right )} c^{2} d + {\left (32 \, C a^{3} b - 25 \, B a^{2} b^{2} + 15 \, A a b^{3}\right )} c d^{2} - 2 \, {\left (24 \, C a^{4} - 20 \, B a^{3} b + 15 \, A a^{2} b^{2}\right )} d^{3}\right )} f^{3} + 2 \, {\left (C b^{4} d^{3} x + C a b^{3} d^{3}\right )} e^{3} - {\left ({\left (3 \, C b^{4} c d^{2} - {\left (7 \, C a b^{3} - 5 \, B b^{4}\right )} d^{3}\right )} f x + {\left (3 \, C a b^{3} c d^{2} - {\left (7 \, C a^{2} b^{2} - 5 \, B a b^{3}\right )} d^{3}\right )} f\right )} e^{2} - {\left ({\left (3 \, C b^{4} c^{2} d + 4 \, {\left (7 \, C a b^{3} - 5 \, B b^{4}\right )} c d^{2} - {\left (32 \, C a^{2} b^{2} - 25 \, B a b^{3} + 15 \, A b^{4}\right )} d^{3}\right )} f^{2} x + {\left (3 \, C a b^{3} c^{2} d + 4 \, {\left (7 \, C a^{2} b^{2} - 5 \, B a b^{3}\right )} c d^{2} - {\left (32 \, C a^{3} b - 25 \, B a^{2} b^{2} + 15 \, A a b^{3}\right )} d^{3}\right )} f^{2}\right )} e\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 ({\left (2 \, C b^{4} c^{2} d + {\left (8 \, C a b^{3} - 5 \, B b^{4}\right )} c d^{2} - 2 \, {\left (24 \, C a^{2} b^{2} - 20 \, B a b^{3} + 15 \, A b^{4}\right )} d^{3}\right )} f^{3} x + {\left (2 \, C a b^{3} c^{2} d + {\left (8 \, C a^{2} b^{2} - 5 \, B a b^{3}\right )} c d^{2} - 2 \, {\left (24 \, C a^{3} b - 20 \, B a^{2} b^{2} + 15 \, A a b^{3}\right )} d^{3}\right )} f^{3} + 2 \, {\left (C b^{4} d^{3} f x + C a b^{3} d^{3} f\right )} e^{2} - {\left ({\left (2 \, C b^{4} c d^{2} - {\left (8 \, C a b^{3} - 5 \, B b^{4}\right )} d^{3}\right )} f^{2} x + {\left (2 \, C a b^{3} c d^{2} - {\left (8 \, C a^{2} b^{2} - 5 \, B a b^{3}\right )} d^{3}\right )} f^{2}\right )} e\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 )}}{45 \, {\left (b^{6} d^{3} f^{3} x + a b^{5} d^{3} f^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]