$f(x) = \begin{cases} x^3 + 2x^2 + x + c, & \text{if } x \leq b \\ e^x, & \text{if } x > b. \end{cases}$ Let a function $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined as the piecewise function above.

It is given that $f$ is differentiable for all real values of $x$, and $b$ and $c$ are integers.

Find $b + c$.

**Clarification**: $e$ denotes Euler's number, $e \approx 2.71828$.