\[ \int \frac {\sqrt {d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}\right ) \sqrt {2}}{2 c^{\frac {3}{2}} d^{\frac {3}{2}} e}+\frac {\sqrt {e x +d}}{c d e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}} \]
command
integrate((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right ) e^{\left (-1\right )}}{2 \, \sqrt {-c d} c d} - \frac {\sqrt {2} {\left (\sqrt {c d} \arctan \left (\frac {\sqrt {c d}}{\sqrt {-c d}}\right ) + \sqrt {-c d}\right )} e^{\left (-1\right )}}{2 \, \sqrt {c d} \sqrt {-c d} c d} + \frac {e^{\left (-1\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________