\[\large \int_{{\sqrt{2}} / {\sqrt[8]{3}}}^{\sqrt{2}}\frac{4x^{3}}{x^{8}-4} \, dx = \ln \sqrt{\frac{A+\sqrt{B}}{\sqrt{B}}} \]

The equation above holds true for square-free positive integers \(A\) and \(B\), find \(A+B\).

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