The problem of stability for time-varying delay systems is investigated using a new Lyapunov functional. Two delay-product-type Lyapunov functionals are firstly proposed based on Bessel-Legendre integral inequality and improved free-matrix-based integral inequality. The functional utilizes the information on both the delay $d(t)$ times quadratic function and the delay $h-d(t)$ times quadratic function. Then, by using the Bessel-Legendre integral inequality and the improved reciprocally convex method to estimate the derivative of Lyapunov functional, some less conservative stability criteria are derived for the considered time-varying delay systems. The feasibility and less conservativeness of the obtained criteria are verified by two numerical examples. The results obtained show that the proposed methods can provide a larger admissible delay upper bounds than some existing ones.