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