Minimizing Resultant Volume

Calculus Level pending

Let \(a\), \(b\), \(c\), and, \(d\) be real constants. Minimize the volume of the region bounded between \(y = \dfrac {x^4 + ax^3 +bx^2 + cx + d} {\sqrt[4]{1 - x^2}} \), \(x= -1\) and \(x= 1\), when it is revolved about the \(x\)-axis.

If this volume can be expressed as \(\dfrac mn \pi^2\), where \(m\) and \(n\) are coprime positive integers, submit your answer as \(m+n\).

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