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I have an expression obtain from an integral, which is:

Integrate[ϵ^2/Sqrt[(ϵ - u)^2 + Δ^2], ϵ] // FullSimplify

The denominator and numerator of the result have an obvious common factor:

Sqrt[Δ^2 + (u - ϵ)^2]

How can I let mathematica to cancel this common factor? I have tried Cancel and Refine all the variables to be positive.

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up vote 5 down vote accepted
Integrate[ϵ^2/Sqrt[(ϵ - u)^2 + Δ^2], ϵ] // Apart // FullSimplify
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You could help Mathematica by temporarily renaming the factor.

exp = Integrate[ϵ^2/Sqrt[(ϵ - u)^2 + Δ^2], ϵ] // FullSimplify
exp /. (Δ^2 + (u - ϵ)^2) :> w
FullSimplify[%] /. w :> Δ^2 + (u - ϵ)^2

Mathematica graphics

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