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 $10b+c$ .