A calculus problem by Nishanth Subramanian

Calculus Level 5

25(x2)17(x5)78(x3+9x224x+20)dx\int_2^5 (x-2)^{17}(x-5)^{78}(-x^{3}+9x^{2}-24x+20) \, dx

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

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