An integral with the floor function

Calculus Level 4

$\large \int_1^\infty\frac{\lfloor x\rfloor}{x^3}\, dx = \frac{3}{a}+\frac{\pi b}{12}+\frac{\pi^2 c}{24}-1$

The equation above holds true for where $a,b$ and $c$ are integers $a$ , $b$ and $c$ . Compute the value of $c^{a+b}$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.