Integrate[ (a1 (m+r)+a2(((m+r))^2))r (E^((-(r^2))/(2 (s^2)))),{r,-Infinity,+Infinity}]

Integrate[ (a1 (m+r)+a2(((m+r))^2)) (E^((-(r^2))/(2 (s^2)))),{r,-Infinity,+Infinity}]