Bent Rod Moment

There is a thin uniform rod of mass \(M\) in the \(xy\)-plane which is initially aligned with the \(x\)-axis. The body of the rod begins at the origin \(x=0\) and ends at \(x = L\).

Suppose that the portion of the rod up to \(x = a\), where \(a < L\) remains aligned with the \(x\)-axis, and the portion from \(x = a\) to \(x = L\) is bent upwards at a right angle so as to be perpendicular to the \(xy\)-plane.

The bent rod's moment of inertia with respect to the \(z\)-axis can be expressed as \[{M a^{2} - \dfrac{\alpha}{\beta} \dfrac{ Ma^{3}}{ L}} . \] If \(\alpha\) and \(\beta\) are coprime positive integers, determine \(\alpha+\beta\).

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