First, we can distribute the square root of t inside the parentheses:
$$\int \sqrt{t}\left(t^{2}+t-1\right) dt = \int (t^{\frac{3}{2}} + t^{\frac{1}{2}} - t^{\frac{-1}{2}}) dt$$
Now, we can integrate each term separately:
$$\int (t^{\frac{3}{2}} +
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