A non-ideal battery with constant electromotive force \(\mathcal{E}\) has an internal resistance \(r(t)\) that increases linearly with time due to temperature buildup according to \(r(t) = r_0(1 + \beta t)\), where \(r_0\) and \(\beta\) are positive constants. At time \(t = 0\), the battery is connected to an external load resistor of constant resistance \(R\). Which of the following expressions represents the total energy \(E_R\) dissipated in the load resistor \(R\) from \(t = 0\) to \(t = T\)?