Multiplying through by $x^2$ gives $x^4 + 3x^3 + 3x + 1 = 26x^2$,
or $x^4 + 3x^3 - 26x^2 + 3x + 1 = 0$.
The left-hand side factors as $(x^2 + x - 1)^2 = 0$,
so $x^2 + x - 1 = 0$.
We can solve this quadratic by completing the square or using the quadratic formula,
but we can also recognize that it is the defining equation of the golden ratio $\phi = \frac{1 + \sqrt{5}}{2}$.
Thus, $x = \phi - 1 = \frac{1 + \sqrt{5}}{2} - 1 = \frac{\sqrt{5}}{2}$.
The answer is: 5
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
If $x$ satisfies $x^2 + 3x + \frac{3}x + \frac{1}{x^2} = 26$ and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$.

### Answer:
