Let $u = x + 1$. Then, $x = u - 1$ and $dx = du$. Now, we can rewrite the integral in terms of $u$:
$$\int x e^{(x+1)^{2}} dx = \int (u-1)e^{u^2} du$$
Now, we can use integration by parts. Recall that integration by parts states that:
$$\int u dv = uv - \int v
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