  Fintushel and Stern showed that the Brieskorn sphere $\Sigma(2,3,7)$ bounds a
rational homology ball, while its non-trivial Rokhlin invariant obstructs it
from bounding an integral homology ball. It is known that their argument can be
modified to show that the figure-eight knot is rationally slice, and we use
this fact to provide the first additional examples of Brieskorn spheres that
bound rational homology balls but not integral homology balls: the families
$\Sigma(2,4n+1,12n+5)$ and $\Sigma(3,3n+1,12n+5)$ for $n$ odd. We also provide
handlebody diagrams for a rational homology ball containing a rationally slice
disk for the figure-eight knot, as well as for a rational homology ball bounded
by $\Sigma(2,3,7)$. These handle diagrams necessarily contain 3-handles.
