Inspired by Karthik Sharma

Calculus Level 5

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

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 \(10b+c\).