\[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (e x +d \right )^{1+m} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{e \left (4+m \right )} \]
command
integrate((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (c x^{3} e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right ) + 3\right )} \mathrm {sgn}\left (x e + d\right ) + 3 \, c d x^{2} e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right ) + 2\right )} \mathrm {sgn}\left (x e + d\right ) + 3 \, c d^{2} x e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right ) + 1\right )} \mathrm {sgn}\left (x e + d\right ) + c d^{3} e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right )\right )} \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c}}{m e + 4 \, e} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m}\,{d x} \]________________________________________________________________________________________