\[ \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctan \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {c \left (-b x +a \right )}}\right )}{b \sqrt {c}} \]
command
integrate(1/(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
\[ -\frac {2 \, \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{b \sqrt {-c}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________