A non-linear hardening spring exerts a restoring force \(\vec{F}_s = -(kx + \beta x^3)\hat{i}\), where \(k\) and \(\beta\) are positive constants and \(x\) is the displacement of its free end from equilibrium at \(x = 0\). For an ideal linear spring with \(\beta = 0\), an external agent does work \(W_{\text{ideal}} = \dfrac{3}{2}kd^2\) to slowly compress the spring from \(x = -d\) to \(x = -2d\), where \(d > 0\). Which of the following definite integrals correctly represents the total work done by the external agent to compress the non-linear spring over this same interval from \(x = -d\) to \(x = -2d\)?