If it is given

\[ \int_{-\infty}^\infty\frac{x^2}{x^6 - 2x^5 - 2x^4 + 4x^3 + 3x^2 - 4x + 1} \; dx=\pi, \]

then the value of

\[ \int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1}dx \]

can be expressed as \(\,\dfrac{p\pi}{q}\). Find the value of \(\,p^2+q^2\).

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