Comparison of the remainders of the Simpson quadrature formula and the quadrature formula for three-point rational interpolants

A quadrature formula with positive coefficients is constructed with the use of three nodes a, b, and c = (a + b)/2 and rational interpolants of the form rho(x) = alpha + beta(x - c) + gamma/(x - g) with a pole g determined by nodes outside the integration interval [a, b]. The error of the constructed formula is smaller than the error of the corresponding Simpson quadrature formula if the integrand f (x) has a continuous derivative f((4))(x) on the interval [a, b] and the inequality f(4)(x)f ''(x) > 0 is satisfied.