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Calculus Level 5

$\displaystyle\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n }({ H }_{ n }-\ln { (n+2x) } -\gamma ) } =\dfrac { \gamma }{ a } -\ln { \left( \dfrac { (2x-1)!!\sqrt [ b ]{ \pi } }{ { c }^{ x }x! } \right) }$ If the equation above is true for positive integers $$a$$, $$b$$, $$c$$ and $$x$$, find $$a+b+c$$.

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