Average angle between x and Ax?

Question

Suppose $A\in\mathbb{R}^{n\times n}$ is a fixed positive definite matrix and $x \in \mathbb{R}^n$. The cosine of angle between $x$ and $Ax$ is given by $$ \frac{x^T Ax}{\|x\|\|Ax\|}. $$ I want to know what the average value of the above expression is when $x$ is uniformly distributed on unit sphere in $\mathbb{R}^n$. Diagonalizing $A$ and some algebraic manipulation, one can show that the answer is equal to $$ \int_{\alpha_1^2+\cdots+\alpha_n^2=1} \frac{\sum \lambda_i \alpha_i^2}{\sqrt{\sum\lambda_i^2\alpha_i^2}}\,d \alpha_1\cdots d\alpha_n, $$
where $\lambda_i$, for $1\leq i\leq n$, are the eigenvalues of $A$. However, I do not know how to calculate the above integral.