\[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {4096 d^{4} \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{15015 c e \left (e x +d \right )^{\frac {5}{2}}}-\frac {1024 d^{3} \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{3003 c e \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{13 c e}-\frac {128 d^{2} \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}}}{429 c e \sqrt {e x +d}}-\frac {32 d \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {5}{2}} \sqrt {e x +d}}{143 c e} \]
command
integrate((e*x+d)^(5/2)*(-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}{45045} \, {\left (15015 \, {\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^{5} - 9009 \, {\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^{4} + 858 \, {\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 d^{3} e^{2} + 286 \, {\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^{2} e^{3} - 39 \, {\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 d e^{4} + 5 \, {\left (542 \, \sqrt {2} \sqrt {c d} d^{6} e^{\left (-5\right )} + \frac {{\left (3003 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{5} d^{5} - 9009 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{4} d^{4} - 12870 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{3} d^{3} - 10010 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{2} d^{2} - 4095 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{5} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - 693 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{6} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}\right )} e^{\left (-5\right )}}{c^{6}}\right )} c e^{5}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}\,{d x} \]________________________________________________________________________________________