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This is a bug in SageMath. I've reported it here: https://github.com/sagemath/sage/issues/37946
Here is a workaround, specific to polynomials:
var('x,t')
beta = 7.10000000000000
y = 2.66778786276411e493*t^3*x - 1.93258971534823e247*x
Y = SR(y.polynomial(RR).integral(t))
result = beta*(Y.subs(t=1) - Y.subs(t=0))
