Computing $E(X)$ for a mixed distribution: two atoms plus a continuous piece

Suppose a distribution function $F:\mathbb{R}\to[0,1]$ of a random variable X is as follows: $F(x)=0$ if $x<-2$; $1/3$ if $-2\le x<0$; $1/2$ if $0\le x<5$; $1/2+(x-5)^2/2$ if $5\le x<6$; $1$ if $x\ge6$. Then $E(X)$ is: