\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (2 a C d f -b \left (7 B d f -6 C \left (c f +d e \right )\right )\right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, \sqrt {f x +e}}{35 b \,d^{2} f^{2}}+\frac {2 C \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}\, \sqrt {f x +e}}{7 b d f}-\frac {2 \left (5 b d f \left (-7 A b d f +a c C f +a C d e +5 b c C e \right )+\left (3 a d f -4 b \left (c f +d e \right )\right ) \left (2 a C d f -b \left (7 B d f -6 C \left (c f +d e \right )\right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{105 b \,d^{3} f^{3}}-\frac {2 \left (3 b d f \left (5 a d f \left (-7 A b d f +a c C f +a C d e +5 b c C e \right )-\left (a c f +a d e +3 b c e \right ) \left (2 a C d f -b \left (7 B d f -6 C \left (c f +d e \right )\right )\right )\right )+2 \left (\frac {a d f}{2}-b \left (c f +d e \right )\right ) \left (5 b d f \left (-7 A b d f +a c C f +a C d e +5 b c C e \right )+\left (3 a d f -4 b \left (c f +d e \right )\right ) \left (2 a C d f -b \left (7 B d f -6 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^{2} 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 (3 a^{2} C \,d^{2} f^{2} \left (-c f +d e \right )-3 a b d f \left (7 d f \left (-5 A d f +2 B c f +3 B d e \right )-C \left (11 c^{2} f^{2}+8 c d e f +16 d^{2} e^{2}\right )\right )-b^{2} \left (C \left (24 c^{3} f^{3}+17 c^{2} d e \,f^{2}+16 c \,d^{2} e^{2} f +48 d^{3} e^{3}\right )+7 d f \left (5 A d f \left (c f +2 d e \right )-B \left (4 c^{2} f^{2}+3 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}}}{105 b^{2} d^{\frac {7}{2}} f^{4} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((b*x+a)^(3/2)*(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 (6 \, C b^{4} c d^{3} - {\left (8 \, C a b^{3} + 7 \, B b^{4}\right )} d^{4}\right )} f^{4} x + {\left (24 \, C b^{4} c^{2} d^{2} - {\left (33 \, C a b^{3} + 28 \, B b^{4}\right )} c d^{3} + {\left (3 \, C a^{2} b^{2} + 42 \, 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 (23 \, C b^{4} c d^{3} - {\left (33 \, 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 (48 \, C b^{4} c^{4} - 8 \, {\left (12 \, C a b^{3} + 7 \, B b^{4}\right )} c^{3} d + {\left (39 \, C a^{2} b^{2} + 119 \, B a b^{3} + 70 \, A b^{4}\right )} c^{2} d^{2} + {\left (9 \, C a^{3} b - 56 \, B a^{2} b^{2} - 175 \, A a b^{3}\right )} c d^{3} + {\left (6 \, C a^{4} - 21 \, B a^{3} b + 175 \, A a^{2} b^{2}\right )} d^{4}\right )} f^{4} + {\left (16 \, C b^{4} c^{3} d - 7 \, {\left (4 \, C a b^{3} + 3 \, B b^{4}\right )} c^{2} d^{2} + 7 \, {\left (C a^{2} b^{2} + 7 \, B a b^{3} + 5 \, A b^{4}\right )} c d^{3} + {\left (9 \, C a^{3} b - 56 \, B a^{2} b^{2} - 175 \, A a b^{3}\right )} d^{4}\right )} f^{3} e + {\left (11 \, C b^{4} c^{2} d^{2} - 7 \, {\left (4 \, C a b^{3} + 3 \, B b^{4}\right )} c d^{3} + {\left (39 \, C a^{2} b^{2} + 119 \, B a b^{3} + 70 \, A b^{4}\right )} d^{4}\right )} f^{2} e^{2} + 8 \, {\left (2 \, C b^{4} c d^{3} - {\left (12 \, 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 (48 \, C b^{4} c^{3} d - 8 \, {\left (9 \, C a b^{3} + 7 \, B b^{4}\right )} c^{2} d^{2} + {\left (12 \, C a^{2} b^{2} + 91 \, B a b^{3} + 70 \, A b^{4}\right )} c d^{3} + {\left (6 \, C a^{3} b - 21 \, B a^{2} b^{2} - 140 \, A a b^{3}\right )} d^{4}\right )} f^{4} + {\left (40 \, C b^{4} c^{2} d^{2} - {\left (62 \, C a b^{3} + 49 \, B b^{4}\right )} c d^{3} + {\left (12 \, C a^{2} b^{2} + 91 \, B a b^{3} + 70 \, A b^{4}\right )} d^{4}\right )} f^{3} e + 8 \, {\left (5 \, C b^{4} c d^{3} - {\left (9 \, 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^{3} d^{5} f^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C b x^{3} + {\left (C a + B b\right )} x^{2} + A a + {\left (B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{d f x^{2} + c e + {\left (d e + c f\right )} x}, x\right ) \]