Let \(E_1\) be a normed algebra with a unit element, \(E_2\) be a Banach space and let \(f:E_1\rightarrow E_2\). In the paper the Hyers-Ulam-Rassias stability of the Davison functional equation \[ f(xy)+f(x+y)=f(xy+x)+f(y) \] is proved. As a consequence of the main theorem the authors obtain among others the following: Let \(\varepsilon\geq 0\) and \(p\in (0,1)\). If \(f\) satisfies \[ \left\|f(xy)+f(x+y)-f(xy+x)-f(y)\right\|\leq\varepsilon\left(\|x\|^p+\|y\|^p\right) \] for all \(x,y\in E_1\), then there exists a unique additive function \(A:E_1\rightarrow E_2\) such that \[ \left\|f(x)-A(x)-f(0)\right\|\leq 4\varepsilon+ \left(1+2^{1-p}+{3^{1-p}(2^p+1)\over 2^p(2^{1+p}-1)}\right) \varepsilon\|x\|^p \] for all \(x\in E_1\).