A brilliant sum

Calculus Level 5

\[\alpha= \displaystyle \sum_{b=1}^{\infty}\left(\displaystyle \sum_{r=1}^{\infty}\left(\displaystyle \sum_{i=1}^{\infty}\left(\displaystyle \sum_{l=1}^{\infty}\left(\displaystyle \sum_{a=1}^{\infty}\left(\displaystyle \sum_{n=1}^{\infty}\left(\displaystyle \sum_{t=1}^{\infty}\left(\dfrac{1}{x^{b+r+i+l+l+i+a+n+t}}\right)\right)\right)\right)\right)\right)\right)\]

If \(\alpha=\dfrac{1}{(x+a)^b(x+c)^d}\) find \(|a|+|b|+|c|+|d|\)

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