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$\int_2^5 (x-2)^{17}(x-5)^{78}(-x^{3}+9x^{2}-24x+20) \, dx$

If the above integral can be expressed as $\dfrac{a^b}{b!} c!d!$, where $b$,$c$ and $d$ are mutually coprime integers, find $a+b+c+d$.

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