\[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{3 b \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {2 \left (2 a^{3} C d f +a \,b^{2} \left (-4 A d f +B c f +B d e +6 c C e \right )-b^{3} \left (3 B c e -2 A \left (c f +d e \right )\right )+a^{2} b \left (B d f -4 C \left (c f +d e \right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{3 b \left (-a d +b c \right )^{2} \left (-a f +b e \right )^{2} \sqrt {b x +a}}-\frac {2 \left (2 a^{3} C d f +a \,b^{2} \left (-4 A d f +B c f +B d e +6 c C e \right )-b^{3} \left (3 B c e -2 A \left (c f +d e \right )\right )+a^{2} b \left (B d f -4 C \left (c f +d e \right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {d}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{3 b^{2} \left (a d -b c \right )^{\frac {3}{2}} \left (-a f +b e \right )^{2} \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}-\frac {2 \left (a^{2} C d \left (-c f +d e \right )-b^{2} \left (A c d f +2 A \,d^{2} e -3 B c d e +3 c^{2} C e \right )+a b \left (3 \left (A \,d^{2}+c^{2} C \right ) f -B d \left (2 c f +d e \right )\right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{3 b^{2} \left (a d -b c \right )^{\frac {3}{2}} \left (-a f +b e \right ) \sqrt {d}\, \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)/(b*x+a)^(5/2)/(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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{3} d f x^{5} + a^{3} c e + {\left (b^{3} d e + {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} x^{4} + {\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f\right )} x^{3} + {\left (3 \, {\left (a b^{2} c + a^{2} b d\right )} e + {\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} x^{2} + {\left (a^{3} c f + {\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} x}, x\right ) \]