\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ \frac {2 C \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \sqrt {f x +e}}{7 b d f}-\frac {2 \left (4 a C d f +b \left (-7 B d f +4 c C f +6 C d e \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}\, \sqrt {f x +e}}{35 b \,d^{2} f^{2}}-\frac {2 \left (5 b d f \left (-7 A b d f +a c C f +3 a C d e +3 b c C e \right )+\left (a d f -2 b \left (c f +2 d e \right )\right ) \left (4 a C d f +b \left (-7 B d f +4 c C f +6 C d e \right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{105 b^{2} d^{2} f^{3}}-\frac {2 \left (3 b d f \left (5 a d f \left (-7 A b d f +a c C f +3 a C d e +3 b c C e \right )-\left (a c f +3 a d e +b c e \right ) \left (4 a C d f +b \left (-7 B d f +4 c C f +6 C d e \right )\right )\right )+2 \left (\frac {b c f}{2}-d \left (a f +b e \right )\right ) \left (5 b d f \left (-7 A b d f +a c C f +3 a C d e +3 b c C e \right )+\left (a d f -2 b \left (c f +2 d e \right )\right ) \left (4 a C d f +b \left (-7 B d f +4 c C f +6 C 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}}{105 b^{3} d^{\frac {5}{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 (4 a^{2} C \,d^{2} f^{2}+a b d f \left (-7 B d f -2 c C f +8 C d e \right )-b^{2} \left (7 d f \left (-10 A d f +B c f +8 B d e \right )-4 C \left (c^{2} f^{2}+2 c d e f +12 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^{3} d^{\frac {5}{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
\[ \frac {2 \, {\left (3 \, {\left (15 \, C b^{4} d^{4} f^{4} x^{2} + 24 \, C b^{4} d^{4} f^{2} e^{2} + 3 \, {\left (C b^{4} c d^{3} + {\left (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 (2 \, C a b^{3} + 7 \, B b^{4}\right )} c d^{3} + {\left (4 \, C a^{2} b^{2} - 7 \, B a b^{3} - 35 \, A b^{4}\right )} d^{4}\right )} f^{4} - {\left (18 \, C b^{4} d^{4} f^{3} x + {\left (5 \, C b^{4} c d^{3} + {\left (5 \, C a b^{3} + 28 \, 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 (48 \, C b^{4} d^{4} e^{4} - {\left (8 \, C b^{4} c^{4} - {\left (9 \, C a b^{3} + 14 \, B b^{4}\right )} c^{3} d - {\left (4 \, C a^{2} b^{2} - 21 \, B a b^{3} - 35 \, A b^{4}\right )} c^{2} d^{2} - {\left (9 \, C a^{3} b - 21 \, B a^{2} b^{2} + 140 \, A a b^{3}\right )} c d^{3} + {\left (8 \, C a^{4} - 14 \, B a^{3} b + 35 \, A a^{2} b^{2}\right )} d^{4}\right )} f^{4} - {\left (5 \, C b^{4} c^{3} d - 7 \, {\left (C a b^{3} + 2 \, B b^{4}\right )} c^{2} d^{2} - 7 \, {\left (C a^{2} b^{2} - 8 \, B a b^{3} - 10 \, A b^{4}\right )} c d^{3} + {\left (5 \, C a^{3} b - 14 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} d^{4}\right )} f^{3} e - {\left (10 \, C b^{4} c^{2} d^{2} - 7 \, {\left (6 \, C a b^{3} + 7 \, B b^{4}\right )} c d^{3} + {\left (10 \, C a^{2} b^{2} - 49 \, B a b^{3} - 70 \, A b^{4}\right )} d^{4}\right )} f^{2} e^{2} - 8 \, {\left (5 \, C b^{4} c d^{3} + {\left (5 \, C a b^{3} + 7 \, 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 (48 \, C b^{4} d^{4} f e^{3} - {\left (8 \, C b^{4} c^{3} d - {\left (5 \, C a b^{3} + 14 \, B b^{4}\right )} c^{2} d^{2} - {\left (5 \, C a^{2} b^{2} - 14 \, B a b^{3} - 35 \, A b^{4}\right )} c d^{3} + {\left (8 \, C a^{3} b - 14 \, B a^{2} b^{2} + 35 \, A a b^{3}\right )} d^{4}\right )} f^{4} - {\left (9 \, C b^{4} c^{2} d^{2} - {\left (8 \, C a b^{3} + 21 \, B b^{4}\right )} c d^{3} + {\left (9 \, C a^{2} b^{2} - 21 \, B a b^{3} - 70 \, A b^{4}\right )} d^{4}\right )} f^{3} e - 8 \, {\left (2 \, C b^{4} c d^{3} + {\left (2 \, C a b^{3} + 7 \, 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^{4} d^{4} f^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c}}{\sqrt {f x + e}}, x\right ) \]