Is There a Computer Algebra System That Can Handle Big O Error Terms?

[CRGreathouse]

I was wondering if there was a computer algebra system (like Math'ca, Maple, or Maxima) that could handle manipulations with big O error terms.

Mathematica has an O function, but it doesn't work properly:

> x^2 Log[x] + x Log[x]^2 + x / Log[x] + O[x]

O[x^1]

(it should be "x^2 Log[x] + x Log[x]^2 + O[x]")

But I'm interested in more than just cutting off small terms. I'd like to be able to do division

> (x^2 + 3)/(x^2 + O(x))

1 + O(1/x)

and harder functions like inverses and integrals

> inverse(x^2 + O(x), x)

±sqrt(x) + O(1)

(Note: the Os in my examples are about +infinity; a system handling other limits would be nice.)

Is there any system that can do this? If not, is there a reason? Is it harder than it seems?