\[ \int \frac {(d+e x)^{7/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {24 d \left (e x +d \right )^{\frac {3}{2}} \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{35 c e}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{7 c e}-\frac {256 d^{3} \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{35 c e \sqrt {e x +d}}-\frac {64 d^{2} \sqrt {e x +d}\, \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{35 c e} \]
command
integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{35} \, {\left (\frac {128 \, \sqrt {2} \sqrt {c d} d^{3}}{c} - \frac {280 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} d^{3}}{c} + \frac {140 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2} d^{2} - 42 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c d - 5 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{4}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {-c e^{2} x^{2} + c d^{2}}}\,{d x} \]________________________________________________________________________________________