# Insane Integral

Calculus Level 5

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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