\[\displaystyle\int_0^1 \frac{x^3}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,\mathrm dx=\dfrac{a-b\sqrt{b}}{c}\]

Where \(a,b\) are coprime and \(c\) is multiple of both \(b,a\).Find \(a+b+c\).

Clarification: \(a,b,c\) are all integers.

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