Step 1: Find the common multiples:
Numbers divisible by both 4 and 5 are exactly the multiples of \(\text{LCM}(4,5) = 20\). Between 1 and 400 these are \(20 \times 1, 20 \times 2, \ldots, 20 \times 19\), because \(20 \times 20 = 400\) is not between 1 and 400.
Step 2: Count them:
Dividing 399 by 20 gives 19 with a remainder, so there are 19 such multiples.
Step 3: Take 20 out as a common factor:
\[ S = 20(1 + 2 + 3 + \cdots + 19) \]
Step 4: Use the sum of the first 19 natural numbers:
\[ 1 + 2 + \cdots + 19 = \frac{19 \times 20}{2} = 190 \]
\[ S = 20 \times 190 = 3800 \]
Check with the A.P. formula: \(\dfrac{19}{2}(20 + 380) = 3800\).
Final Answer:
The sum of all such integers is 3800.
\[ \boxed{3800} \]