[Mathematica] Bug in Integrate with derivatives of a delta function

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SUMMARY

The forum discussion addresses a bug in Mathematica 8 for Windows related to the integration of expressions involving the derivative of the Dirac delta function. Specifically, the integrals Integrate[f[qs] DiracDelta'[qs (1 - 1/x)], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] and Integrate[f[qs] DiracDelta'[qs - qs/x], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] yield results that differ by a sign, despite being mathematically equivalent. The issue appears to be linked to the representation of the Dirac delta function's argument, particularly when involving the first negative power of x.

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Code: 
Integrate[f[qs] DiracDelta'[qs (1 - 1/x)], {qs, -\[Infinity], \[Infinity]}, 
 Assumptions -> 0 < x < 1]
Integrate[f[qs] DiracDelta'[qs - qs/x], {qs, -\[Infinity], \[Infinity]}, 
 Assumptions -> 0 < x < 1]

This is on Mathematica 8 for windows.
The results differ by a sign. They are effective the exact same thing, but due to how I write the insides of the dirac delta I get a different result.

Can anyone replicate this?
 
Last edited:
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I notice it is only for the first negative power of x. If you put

[qs (1 - 1/x^(1.))]

[qs - qs/x^(1.)]

it works?
 

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