Upper bound on $\sum_{k=1}^T \frac{1}{k (1+a)^{T-k}}$

Is there any reasonable upper bound for the following quantity $ $ \sum_{k=1}^T \frac{1}{k (1+a)^{T-k}} $ $

where $ a>0$ with respect to $ T$ and $ a$ (something like $ \mathcal{O}(\frac{\log (T)}{aT}$ )? I tried to compute integral $ $ \int_{0}^T \frac{1}{x (1+a)^{T-x}}dx, $ $ which should be upper bound on this sum as $ f(x) = \frac{1}{x (1+a)^{T-x}}$ is decreasing on $ (0, T)$ , but I did not achieve to get reasonable expression.