The interpolating polynomial gives a good approximation taking $x=0$,$x=1/2$ and $x=1$.
$f(x)={{{c}}^b}\left( {x - 1} \right)\left( {2x - 1} \right) - 4x\left( {x - 1} \right){\left( {\frac{{2c + 1}}{{c + 2}}} \right)^b} + x\left( {2x - 1} \right)$
so you can evaluate
$\int_0^1x^{1+b}f(x)dx$
and get an approximation. For $b$ fixed large and $c$ any value, the approximation is very good. Similarily for the other symmetric situation. For $c=0$ and $b<0.1$ the approximation is very bad, but for $c\neq 0$ we can push the values to $.9$ and things look good, for $c<.2;.3$ we get a not too good approximation.
Here you can see some images. $p$ is blue, the original function in red.
Bad $c=0$ case. $b$ ranges $0$ to $1$ in tenths of unity.
Good $b=1$ case. $c$ ranges $0$ to $1$ in tenths of unity.
Not that bad $c=0.1$ case. $b$ ranges $0$ to $1$ in tenths of unity.
Good $c=0.6$ case. $c$ ranges $0$ to $1$ in tenths of unity.
Good $b=0.62$ case. $c$ ranges $0$ to $1$ in tenths of unity.
