The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter
1. **Identify the sides of the triangle and the given angle:**
   The sides of the triangle are \( x \), \( 2x + 1 \), and \( x + 2 \). One of the angles is \( 60^\circ \).

2. **Apply the Law of Cosines:**
   The Law of Cosines states that for any triangle with sides \( a \), \( b \), and \( c \) and an angle \( \gamma \) opposite side \( c \):
   \[
   c^2 = a^2 + b^2 - 2ab \cos(\gamma)
   \]
   Here, we assume the angle \( 60^\circ \) is opposite the side \( x + 2 \). Thus, we have:
   \[
   (x + 2)^2 = x^2 + (2x + 1)^2 - 2 \cdot x \cdot (2x + 1) \cdot \cos(60^\circ)
   \]

3. **Substitute \(\cos(60^\circ) = \frac{1}{2}\):**
   \[
   (x + 2)^2 = x^2 + (2x + 1)^2 - 2 \cdot x \cdot (2x + 1) \cdot \frac{1}{2}
   \]
   Simplify the equation:
   \[
   (x + 2)^2 = x^2 + (2x + 1)^2 - x(2x + 1)
   \]

4. **Expand and simplify the equation:**
   \[
   (x + 2)^2 = x^2 + 4x^2 + 4x + 1 - 2x^2 - x
   \]
   \[
   x^2 + 4x + 4 = 3x^2 + 3x + 1
   \]

5. **Rearrange the equation to form a quadratic equation:**
   \[
   x^2 + 4x + 4 = 3x^2 + 3x + 1
   \]
   \[
   0 = 3x^2 + 3x + 1 - x^2 - 4x - 4
   \]
   \[
   0 = 2x^2 - x - 3
   \]

6. **Solve the quadratic equation \( 2x^2 - x - 3 = 0 \):**
   Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -1 \), and \( c = -3 \):
   \[
   x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}
   \]
   \[
   x = \frac{1 \pm \sqrt{1 + 24}}{4}
   \]
   \[
   x = \frac{1 \pm \sqrt{25}}{4}
   \]
   \[
   x = \frac{1 \pm 5}{4}
   \]
   \[
   x = \frac{6}{4} \quad \text{or} \quad x = \frac{-4}{4}
   \]
   \[
   x = \frac{3}{2} \quad \text{or} \quad x = -1
   \]
   Since \( x \) must be positive, we have \( x = \frac{3}{2} \).

7. **Calculate the perimeter of the triangle:**
   \[
   \text{Perimeter} = x + (2x + 1) + (x + 2)
   \]
   \[
   \text{Perimeter} = \frac{3}{2} + 2 \cdot \frac{3}{2} + 1 + \frac{3}{2} + 2
   \]
   \[
   \text{Perimeter} = \frac{3}{2} + 3 + 1 + \frac{3}{2} + 2
   \]
   \[
   \text{Perimeter} = \frac{3}{2} + \frac{3}{2} + 3 + 1 + 2
   \]
   \[
   \text{Perimeter} = 3 + 3 + 3
   \]
   \[
   \text{Perimeter} = 9
   \]

The final answer is \(\boxed{9}\).