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