\[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {16 d \left (e x +d \right )^{\frac {3}{2}}}{3 c e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}-\frac {2 \left (e x +d \right )^{\frac {5}{2}}}{3 c e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}+\frac {64 d^{2} \sqrt {e x +d}}{3 c e \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}} \]
command
integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {32 \, \sqrt {2} d^{2} e^{\left (-1\right )}}{3 \, \sqrt {c d} c} + \frac {8 \, d^{2} e^{\left (-1\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d} c} + \frac {2 \, {\left (12 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{7} d e^{2} - {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{6} e^{2}\right )} e^{\left (-3\right )}}{3 \, c^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________