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