To solve the equation \( 6^{3x+1} = 36^{x+3} \), we can start by rewriting \( 36 \) in terms of base \( 6 \):

\[
36 = 6^2
\]

Therefore, we can express \( 36^{x+3} \) as:

\[
36^{x+3} = (6^2)^{x+3} = 6^{2(x+3)} = 6^{2x + 6}
\]

Now our equation becomes:

\[
6^{3x + 1} = 6^{2x + 6}
\]

Since the bases are the same, we can set the exponents equal to each other:

\[
3x + 1 = 2x + 6
\]

Next, we can solve for \( x \):

1. Subtract \( 2x \) from both sides:
\[
3x - 2x + 1 = 6
\]
This simplifies to:
\[
x + 1 = 6
\]

2. Subtract \( 1 \) from both sides:
\[
x = 5
\]

Thus, the solution to the equation \( 6^{3x+1} = 36^{x+3} \) is:

\[
\boxed{5}
\]