$\large I = \int_{0}^{\infty} \dfrac{x^{4}}{(1+x^{4})^{2}} \, dx$

If the value of the integral above is of the form $\dfrac{\pi}{b\sqrt{c}}$, where $b$, $c$ are integers and $c$ is square-free, find the value of $10b+c$.

This is my original problem.

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