Reciprocal Sum Equations!

Algebra Level 5

\[\large{\dfrac{1}{x} + \dfrac{1}{x+2} - \dfrac{1}{x+4} - \dfrac{1}{x+6} - \dfrac{1}{x+8} - \dfrac{1}{x+10} + \dfrac{1}{x+12} + \dfrac{1}{x+14} = 0}\]

Given that \( \{A,B,C,D,E \}\) are the only five distinct real values of \(x\) satisfying the above equation such that \(A\) is an integer, and \(B,C,D,E\) are irrational numbers of the form \(-\alpha \pm \sqrt{\beta \pm \gamma \sqrt{\delta}}\), where \(\alpha, \beta, \gamma, \delta\) are positive integers with \(\beta\) and \(\delta\) being prime numbers.

Submit the value of \(A+\alpha+\beta+\gamma+\delta\) as your answer.

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