I can! Can you? (2)

Calculus Level 5

If

\[\int_0^{\infty}\frac{1}{2x^4+x^2+1}dx=\pi\sqrt{\frac{1}{a\sqrt{b}}-\frac{1}{c}}\]

where \(b\) is square-free, what is the value of \(a+b+c?\)

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