\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {8 d \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{35 c e \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{7 c e \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{105} \, {\left (35 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} c d^{2} - {\left (22 \, \sqrt {2} \sqrt {c d} d^{3} e^{\left (-2\right )} - \frac {{\left (35 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2} d^{2} - 42 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - 15 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}\right )} e^{\left (-2\right )}}{c^{3}}\right )} c e^{2}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{\sqrt {e x + d}}\,{d x} \]________________________________________________________________________________________