I am trying to evaluate the following two beta function integrals
$$ \sum_{k=0}^\infty x^k B(k,5/2) = \int ^1_0 \left[\frac{(1-t)^{3/2}}{t}\right]\frac{1}{1-xt}dt $$
and
$$ \sum_{k=1}^\infty x^k B(k+1,3/2) = \int ^1_0 t\sqrt{1-t}\frac{1}{1-xt}dt $$
How can I proceed?
My attempt at the second integral. With $u = \sqrt{1-t}$
$$ \int ^1_0 t\sqrt{1-t}\frac{1}{1-xt}dt= -2\int^1_0\frac{u^2(1-u^2)}{1-x(1-u^2)}du $$
My attempt at the first integral with again $u=\sqrt{1-t}$ is
$$ \int ^1_0 \left[\frac{(1-t)^{3/2}}{t}\right]\frac{1}{1-xt}dt = -2\int^1_0\left[\frac{u^4}{1-u^2}\right]\frac{1}{1-x(1-u^2)}du $$