Trying to mix some concepts!

Calculus Level pending

Consider\[f(x) =4x^{4} -24x^3+31x^2 +6x-8\] be a polynomial function and \(\alpha ,\beta,\gamma, \delta\) are the roots of the equation \( f(x)=0 \). If it is known that sum of any two out of \(\alpha ,\beta ,\gamma ,\delta\) vanishes then compute \[\large \int_{2\alpha}^{2\beta}\dfrac{x^{\delta+1}-6x^{\gamma+1}+2\beta|x|+1}{x^2+4\beta|x|+1} \, dx .\]

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