\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}\right ) \sqrt {2}}{4 d^{\frac {3}{2}} e \sqrt {c}}-\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{2 c d e \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate(1/(e*x+d)^(3/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (\frac {\sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d} d} - \frac {2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{{\left (x e + d\right )} d}\right )} e^{\left (-1\right )}}{4 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________