In this paper we examine quadrature rules $$Qf = \sum\limits_{j = 1}^n {w_j } f(t_j )$$ for the integral $$\int\limits_a^b {w(t)f(t)dt} $$ which are exact for all $$f(t) = t^\lambda (\sqrt {1 + t^2 } )^\mu $$ with ?+μ?d. We specify three distinct families of solutions which have properties not unlike the standard Gauss and Radau quadrature rules. For each integerd the abscissas of the quadrature rules lie within the closed integration interval and are expressed in terms of the zeros of a polynomialq d(y). These polynomialsq d(y), (d=0, 1, ...), which are not orthogonal, satisfy a three term recurrence relation of the type Qd+1(y)=(y+βd+1)qd(y)??d+1yqd?1(y) and have zeros with the standard interlacing property.