$\displaystyle \int_{0}^{12} \Bigl( \{x\}^2+\lfloor x \rfloor ^2\Bigr) \ dx = \ ?$

**Details and assumptions**:

- Every $x\in \mathbb{R}$ can be written as $x=\lfloor x \rfloor + \{x\}$.
- As usual, $\lfloor x \rfloor$ denotes greatest integer less than or equal to $x$.
- $\{x\}$ is the fractional part of $x$.

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