\[ \int \frac {\sqrt {a+b x} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ \frac {2 C \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, \sqrt {f x +e}}{5 b d f}-\frac {2 \left (2 a C d f -b \left (5 B d f -4 C \left (c f +d e \right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{15 b \,d^{2} f^{2}}-\frac {2 \left (3 b d f \left (-5 A b d f +a c C f +a C d e +3 b c C e \right )+\left (a d f -2 b \left (c f +d e \right )\right ) \left (2 a C d f -b \left (5 B d f -4 C \left (c f +d e \right )\right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{15 b^{2} d^{\frac {5}{2}} f^{3} \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}-\frac {2 \left (-a f +b e \right ) \left (a C d f \left (-c f +d e \right )-b \left (5 d f \left (-3 A d f +B c f +2 B d e \right )-C \left (4 c^{2} f^{2}+3 c d e f +8 d^{2} e^{2}\right )\right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{15 b^{2} d^{\frac {5}{2}} f^{3} \sqrt {d x +c}\, \sqrt {f x +e}} \]
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
\[ \frac {2 \, {\left (3 \, {\left (3 \, C b^{3} d^{3} f^{3} x - 4 \, C b^{3} d^{3} f^{2} e - {\left (4 \, C b^{3} c d^{2} - {\left (C a b^{2} + 5 \, B b^{3}\right )} d^{3}\right )} f^{3}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} - {\left (8 \, C b^{3} d^{3} e^{3} + {\left (8 \, C b^{3} c^{3} - {\left (7 \, C a b^{2} + 10 \, B b^{3}\right )} c^{2} d - {\left (2 \, C a^{2} b - 10 \, B a b^{2} - 15 \, A b^{3}\right )} c d^{2} - {\left (2 \, C a^{3} - 5 \, B a^{2} b + 30 \, A a b^{2}\right )} d^{3}\right )} f^{3} + {\left (3 \, C b^{3} c^{2} d - {\left (2 \, C a b^{2} + 5 \, B b^{3}\right )} c d^{2} - {\left (2 \, C a^{2} b - 10 \, B a b^{2} - 15 \, A b^{3}\right )} d^{3}\right )} f^{2} e + {\left (3 \, C b^{3} c d^{2} - {\left (7 \, C a b^{2} + 10 \, B b^{3}\right )} d^{3}\right )} f e^{2}\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^{3} d^{3} f e^{2} + {\left (8 \, C b^{3} c^{2} d - {\left (3 \, C a b^{2} + 10 \, B b^{3}\right )} c d^{2} - {\left (2 \, C a^{2} b - 5 \, B a b^{2} - 15 \, A b^{3}\right )} d^{3}\right )} f^{3} + {\left (7 \, C b^{3} c d^{2} - {\left (3 \, C a b^{2} + 10 \, B b^{3}\right )} d^{3}\right )} f^{2} 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 \, b^{3} d^{4} f^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{d f x^{2} + c e + {\left (d e + c f\right )} x}, x\right ) \]