\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {f x +e}}{3 b \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (4 a^{2} C f +b^{2} \left (-2 A f +3 B e \right )-a b \left (B f +6 C e \right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{3 b^{2} \left (-a f +b e \right )^{2} \sqrt {b x +a}}+\frac {2 \left (8 a^{3} C d \,f^{2}-a^{2} b f \left (2 B d f +7 c C f +13 C d e \right )+a \,b^{2} \left (3 C e \left (4 c f +d e \right )+f \left (-A d f +B c f +4 B d e \right )\right )-b^{3} \left (A d e f +c \left (-2 A \,f^{2}+3 B e f +3 C \,e^{2}\right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {d}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{3 b^{3} f \left (-a f +b e \right )^{2} \sqrt {a d -b c}\, \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}+\frac {2 \left (-c f +d e \right ) \left (4 a^{2} C d f +b^{2} \left (A d f +3 c C e \right )-a b \left (B d f +3 C \left (c f +d e \right )\right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{3 b^{3} f \left (-a f +b e \right ) \sqrt {d}\, \sqrt {a d -b c}\, \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{3} f x^{4} + a^{3} e + {\left (b^{3} e + 3 \, a b^{2} f\right )} x^{3} + 3 \, {\left (a b^{2} e + a^{2} b f\right )} x^{2} + {\left (3 \, a^{2} b e + a^{3} f\right )} x}, x\right ) \]