To solve the given problem, follow these steps:

1. **Calculate the value of \( A \) when \( x = 36 \):**

\( A = \frac{\sqrt{x} + 3}{\sqrt{x}} = 1 + \frac{3}{\sqrt{x}} \)

Substitute \( x = 36 \):

\[
A = 1 + \frac{3}{\sqrt{36}} = 1 + \frac{3}{6} = 1 + 0.5 = 1.5
\]

2. **Simplify the expression for \( B \):**

\[
B = \left( \frac{x+3}{x-9} + \frac{1}{\sqrt{x}+3} \right) \frac{\sqrt{x} - 3}{\sqrt{x}}
\]

Simplify each part of the expression, focusing on combining fractions and rationalizing the denominator if necessary.

3. **Find all integer values of \( x \) such that \( 2M > \sqrt{5M} \) where \( M = A \cdot B \):**

Substitute \( A = 1.5 \) into \( M = A \cdot B \), and simplify the inequality:

\[
2M = 3B > \sqrt{5M}
\]

This inequality can be solved by substituting possible integer values for \( x \) and checking the condition.

Once each of these steps is completed, the problem should be fully solved. Note that without simplification of \( B \), we need to analyze specific values for \( x \) to satisfy the inequality.