I think something must be getting lost in the translation here.
Out= -3.2224094*^-6 + 4.69189*^-7*x - 8.04567232*^-17*x^4
In:= Integrate [%,x]
Out= -3.2224094*^-6*x + 2.345945*^-7*x^2 - 1.6091344640000002*^-17*x^5
Since that is so different from your integration I have to think there must be some error in the transmission. If you can check the details and clarify the explanation of the problem then perhaps we can get you where you need to go.
The Simplify step isn't actually necessary. I first just did the integration, saw the result was so different and then went back and did the simplification to see if I had missed something. Your input appears to be a simple polynomial in disguise.