Integrate It to Infinity

Calculus Level 5

\[\large \int_0^\infty\dfrac{x^2 }{(x^2+a^2)(x^2+b^2)(x^2+c^2)} \, dx =\dfrac{\pi}{n}\]

Let \(a,b\) and \(c\) be the real roots of the cubic equation \(x^3-7x^2+10x-2=0\). Find the value of the positive integer \(n\) that satisfy the equation above.

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