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