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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