By using the function values at three nodes in the integration interval and the 1st and 2nd-order derivative values, a numerical integration formula with 9-degree algebraic accuracy has thus been constructed firstly in this current research. Then, a composite formula and acceleration formula are provided for this proposed formula, followed by an error analysis for each new numerical integration formula, with (O(h10)) the given convergence order of the accelerated numerical integration formula. Finally, the high efficiency of these formulas is verified through several numerical examples.