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Given a hyperelliptic plane curve $$ X = \text{Spec}\left( \frac{\mathbb{C}[x,y]}{y^2 - x(x-1)(x-2)^4} \right) $$ how can I find its normalization?

I know I am suppose to compute the integral closure of the morphism $$ \frac{\mathbb{C}[x,y]}{y^2 - x(x-1)(x-2)^4} \to \text{Frac}\left( \frac{\mathbb{C}[x,y]}{y^2 - x(x-1)(x-2)^4} \right) $$ and this will give me the algebra of the normalized curve.

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    See here: http://math.stackexchange.com/a/681926/121097 – 2017-02-06
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    @user26857 the target is to get a Dedekind domain whose fraction field is the function field of the curve ? – 2017-02-06

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