Integral of a defined function-

Calculus Level 4

Let a function $f(x)$ be defined as:

$f(x) = \displaystyle\sum_{r=0}^x r^{2}~\dbinom{x}{r}.$

Find the value of

$24 \displaystyle\int_0^{1} \dfrac{f(x)}{2^{x}} dx.$

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