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This doesn't fix the underlying issue that Maxima really doesn't like floating point exponents for some reason (see here for a related issue), but since you probably mean exact precision with your exponent here, try this:

sage: (x*(0.6*x^(1/2)+0.6)).integrate(x,0,1)
0.54

The underlying question is whether x^0.5 or x to any decimal power (as a "floating point" object) is really the same as x^(1/2), and if it isn't, then what the heck does it mean? These are subtle questions that different systems handle differently.