\[ \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 \arctanh \left (\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}\right ) \sqrt {2}}{8 c^{\frac {3}{2}} d^{\frac {5}{2}} e}-\frac {1}{2 c d e \sqrt {e x +d}\, \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}+\frac {3 \sqrt {e x +d}}{4 c \,d^{2} e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}} \]
command
integrate(1/(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
\[ \frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right ) e^{\left (-1\right )}}{8 \, \sqrt {-c d} c d^{2}} + \frac {{\left (3 \, {\left (x e + d\right )} c - 2 \, c d\right )} e^{\left (-1\right )}}{4 \, {\left (2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}\right )} c d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________