Let $u = x + 1$, so $x = u - 1$ and $d x = d u$. The limits of integration will also change: when $x = 0$, $u = 1$, and when $x = 4$, $u = 5$. Now we can rewrite the integral as:
$$\int_{1}^{5} \frac{(u-1)^{3/2}}{u} d u$$
Now, let's use integration by parts. Let
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