$\displaystyle \int \int_D \frac{dxdy}{\sqrt{xy}}$ where $\displaystyle D = \left\{x+y\leq1, x> 0, y> 0\right\}$

$\displaystyle \int_b^1 \left(\int_b^{1-x} \frac{dy}{\sqrt{xy}}\right)dx = 2\int_b^1 \frac{1}{x}(\sqrt{x-x^2} -\sqrt{bx})dx$

I can't solve the last integral, and the answer I get from Mathematica does not work for these limits. So I suspect the above isn't correct.

EDIT: Here's another tricky improper integral:

$\displaystyle \int \int \int_D \frac{dxdydz}{\sqrt{x^2+y^2+z^2}}$ where $\displaystyle \sqrt{x^2+y^2} < z < 1$

EDIT: And another one:

$\displaystyle \int \int_{R^2} \frac{xdxdy}{(x^2+y^2)^2 +1}$