\[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{c e \sqrt {e x +d}} \]
command
integrate((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
\[ 2 \, {\left (\frac {\sqrt {2} \sqrt {c d}}{c} - \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {e x + d}}{\sqrt {-c e^{2} x^{2} + c d^{2}}}\,{d x} \]________________________________________________________________________________________