Viva La Revolucion!

Calculus Level 3

The volume of the solid formed by revolving the curve

$y= \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+1}}$

bounded between $x = \frac{1}{8}$ and $x=\frac{9}{16}$ around the $x$ -axis is in the the form $A \pi \ln{\frac{B}{C}}.$

If $A$ and $B$ are square-free, what is the value of $A \times B \times C?$