\[ \int \frac {1}{\sqrt {a+b x} \sqrt {-a d+b d x}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {b d x -a d}}\right )}{b \sqrt {d}} \]
command
integrate(1/(b*x+a)^(1/2)/(b*d*x-a*d)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \log \left ({\left | -\sqrt {b x + a} \sqrt {d} + \sqrt {{\left (b x + a\right )} d - 2 \, a d} \right |}\right )}{b \sqrt {d}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________