jmc

The given function has a domain of all real numbers if and only if the denominator is never equal to zero. In other words, the quadratic $x^2+4x+c=0$ has no real roots. The discriminant of this quadratic is $16-4c$. The quadratic has no real roots if and only if the discriminant is negative, so $16-4c<0$, or $c>4$. The smallest integer $c$ that satisfies this inequality is $c=\boxed{5}$.