\[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {256 d^{3} \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{1155 c e \left (e x +d \right )^{\frac {5}{2}}}-\frac {64 d^{2} \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{231 c e \left (e x +d \right )^{\frac {3}{2}}}-\frac {8 d \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{33 c e \sqrt {e x +d}}-\frac {2 \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}} \sqrt {e x +d}}{11 c e} \]
command
integrate((e*x+d)^(3/2)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{3465} \, {\left (1155 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} c d^{4} - 462 \, {\left (2 \, \sqrt {2} \sqrt {c d} d^{2} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{2}}\right )} c d^{3} + 22 \, {\left (26 \, \sqrt {2} \sqrt {c d} d^{4} e^{\left (-3\right )} + \frac {{\left (105 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{3} d^{3} - 189 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{2} d^{2} - 135 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - 35 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}\right )} e^{\left (-3\right )}}{c^{4}}\right )} c d e^{3} - {\left (422 \, \sqrt {2} \sqrt {c d} d^{5} e^{\left (-4\right )} - \frac {{\left (1155 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{4} d^{4} - 2772 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{3} d^{3} - 2970 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{2} d^{2} - 1540 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - 315 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{5} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}\right )} e^{\left (-4\right )}}{c^{5}}\right )} c e^{4}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}\,{d x} \]________________________________________________________________________________________