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Algebra Level 5

$\dfrac{5^3}{4(3!+4!+5!)}-\dfrac{6^3}{5(4!+5!+6!)}+\cdots+\dfrac{2017^3}{2016(2015!+2016!+2017!)}$

If the sum of above series can be written as $$\dfrac{1}{a!}+\dfrac{1}{b!}+e^c+d$$ for integers $$a, b, c, d$$, then submit $$a+b+c+d$$.

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