$a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$ be differentiable on $\mathbb{R}$. Then, the value of $\int_\limits{-2}^2 f(x) d x$ equals