\[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx \]
Optimal antiderivative \[ \frac {f \left (A +\frac {e \left (-B f +C e \right )}{f^{2}}\right ) \left (-b^{2} x^{2}+a^{2}\right )}{\left (-a^{2} f^{2}+b^{2} e^{2}\right ) \left (f x +e \right ) \sqrt {b x +a}\, \sqrt {-b c x +a c}}+\frac {C \arctan \left (\frac {b x \sqrt {c}}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{b \,f^{2} \sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b c x +a c}}+\frac {\left (a^{2} f^{2} \left (-B f +2 C e \right )-b^{2} \left (-A e \,f^{2}+C \,e^{3}\right )\right ) \arctan \left (\frac {\left (b^{2} e x +a^{2} f \right ) \sqrt {c}}{\sqrt {-a^{2} f^{2}+b^{2} e^{2}}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{f^{2} \left (-a^{2} f^{2}+b^{2} e^{2}\right )^{\frac {3}{2}} \sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b c x +a c}} \]
command
integrate((C*x^2+B*x+A)/(f*x+e)^2/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {2 \, {\left (B a^{2} b \sqrt {-c} f^{3} - 2 \, C a^{2} b \sqrt {-c} f^{2} e - A b^{3} \sqrt {-c} f^{2} e + C b^{3} \sqrt {-c} e^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} f - 2 \, b c e}{2 \, \sqrt {a^{2} f^{2} - b^{2} e^{2}} c}\right )}{{\left (a^{2} f^{4} - b^{2} f^{2} e^{2}\right )} \sqrt {a^{2} f^{2} - b^{2} e^{2}} c} + \frac {C \log \left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}\right )}{\sqrt {-c} f^{2}} + \frac {4 \, {\left (2 \, A a^{2} b^{2} \sqrt {-c} c f^{3} - A b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} f^{2} e - 2 \, B a^{2} b^{2} \sqrt {-c} c f^{2} e + B b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} f e^{2} + 2 \, C a^{2} b^{2} \sqrt {-c} c f e^{2} - C b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e^{3}\right )}}{{\left (a^{2} f^{4} - b^{2} f^{2} e^{2}\right )} {\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} f + 4 \, a^{2} c^{2} f - 4 \, b {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c e\right )}}}{b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________