\[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx \]
Optimal antiderivative \[ -\frac {B \left (-b^{2} x^{2}+a^{2}\right )}{b^{2} \sqrt {b x +a}\, \sqrt {-b c x +a c}}-\frac {C x \left (-b^{2} x^{2}+a^{2}\right )}{2 b^{2} \sqrt {b x +a}\, \sqrt {-b c x +a c}}+\frac {\left (2 A \,b^{2}+a^{2} C \right ) \arctan \left (\frac {b x \sqrt {c}}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{2 b^{3} \sqrt {c}\, \sqrt {b x +a}\, \sqrt {-b c x +a c}} \]
command
integrate((C*x^2+B*x+A)/(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
\[ -\frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} C}{c} - \frac {C a c - 2 \, B b c}{c^{2}}\right )} + \frac {2 \, {\left (C a^{2} + 2 \, A b^{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}}}{2 \, b^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________