\[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {\left (b^{2} \left (3 A \,d^{2} e^{2}-2 c d e \left (-A f +2 B e \right )+c^{2} \left (3 A \,f^{2}-4 B e f +8 C \,e^{2}\right )\right )+a b \left (d^{2} e \left (-8 A f +B e \right )-c^{2} f \left (-B f +8 C e \right )-2 c d \left (4 A \,f^{2}-7 B e f +4 C \,e^{2}\right )\right )+a^{2} \left (C \left (3 c^{2} f^{2}+2 c d e f +3 d^{2} e^{2}\right )+4 d f \left (2 A d f -B \left (c f +d e \right )\right )\right )\right ) \arctanh \left (\frac {\sqrt {-a f +b e}\, \sqrt {d x +c}}{\sqrt {-a d +b c}\, \sqrt {f x +e}}\right )}{4 \left (-a d +b c \right )^{\frac {5}{2}} \left (-a f +b e \right )^{\frac {5}{2}}}-\frac {\left (A \,b^{2}-a \left (b B -a C \right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{2 b \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b x +a \right )^{2}}+\frac {\left (2 a^{3} C d f +a \,b^{2} \left (-6 A d f +B c f +B d e +8 c C e \right )-b^{3} \left (4 B c e -3 A \left (c f +d e \right )\right )+a^{2} b \left (2 B d f -5 C \left (c f +d e \right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{4 b \left (-a d +b c \right )^{2} \left (-a f +b e \right )^{2} \left (b x +a \right )} \]
command
integrate((C*x^2+B*x+A)/(b*x+a)^3/(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 \[ \text {Timed out} \]