Let's let $u = t^4+1$. Then, $du = 4t^3 dt$. Rearranging, we have $dt = \frac{1}{4t^3} du$.
Substituting these values into the integral, we get:
$$\int t^3 \sqrt{t^4+1} dt = \int t^3 \sqrt{u} \cdot \frac{1}{4t^3} du$$
Simplifying, we have:
$$\frac{1}{4} \int
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