\[ \int (d+e x)^m \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx \]
Optimal antiderivative \[ \frac {\left (e x +d \right )^{1+m} \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}}{e \left (2+m \right )} \]
command
integrate((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (x e + d\right )}^{m + 2} \sqrt {c} e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\right )}{m + 2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )}^{m}\,{d x} \]________________________________________________________________________________________