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Calculus Level 4

\[\large\int_{-1}^{1}\{x^2+x-5\} \, dx\]

If the value of the integral above is of the form \(\frac{1}{a}(b\sqrt{c}-d)\) where \(a,b,c,d\) are positive integers and \(b,c\) are primes, find the value of \(a+b+c+d\).

Note: \(\{x\}=\begin{cases}{x-\lfloor{x}\rfloor \quad , \quad x\geq0} \\ { x-\lceil{x}\rceil \quad , \quad x<0} \end{cases}\).

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