Partition this integral!

Calculus Level 2

$\large \int_{1}^{\infty} \dfrac {2x\{x\} - \{x\}^2} {x^2 \lfloor x \rfloor ^2} \, dx$

If the integral above can be expressed as $\dfrac{\pi^a - b} c$ , where $a,b,c$ are all positive integers, find $a+b+c$ .

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