Steps for Python code:

1. Define symbol x.
2. Define the function $A(x) = \frac{\sqrt{3}}{4} (x + x \ln x)$.
3. Integrate $A(x)$ from 1 to 2.
4. Extract p, q, r from the result.
5. Calculate sum.

Let's write the code. [PYTHON] from sympy import symbols, integrate, ln, sqrt, simplify

x = symbols('x')

Side length s(x) = sqrt(x + x*ln(x))

Area A(x) = (sqrt(3)/4) * s(x)^2 = (sqrt(3)/4) * (x + x*ln(x))

We factor out sqrt(3)/4 for the integration part to keep it simple first

integrand = x + x * ln(x) integral_result = integrate(integrand, (x, 1, 2))

print(f"Integral result: {integral_result}")

Now multiply by sqrt(3)/4

volume = (sqrt(3)/4) * integral_result print(f"Volume: {volume}")

We need the form sqrt(3)(p + qln(2))/r

Let's simplify the integral result to see p, q, r

Integral result should be of form (A + B*ln(2))/C

Then Volume = sqrt(3)/4 * (A + Bln(2))/C = sqrt(3)(A + B*ln(2))/(4C)

So p=A, q=B, r=4C (assuming fractions are simplified)

Let's check the value of integral_result

It was 3/4 + 2*ln(2) based on manual calc.

Let's see what sympy gives.

[/PYTHON] ps_output: Integral result: 2log(2) + 3/4 Volume: sqrt(3)(2*log(2) + 3/4)/4

The volume of the solid is given by the integral of the cross-sectional area $A(x)$ from $x=1$ to $x=2$. The side length of the equilateral triangle at position $x$ is $s(x) = y = \sqrt{x + x \ln x}$. The area of an equilateral triangle with side $s$ is $A(x) = \frac{\sqrt{3}}{4} s(x)^2 = \frac{\sqrt{3}}{4} (x + x \ln x)$.

The volume is: [ V = \int_{1}^{2} \frac{\sqrt{3}}{4} (x + x \ln x) , dx = \frac{\sqrt{3}}{4} \int_{1}^{2} (x + x \ln x) , dx ]

From the Python output, the integral $\int_{1}^{2} (x + x \ln x) \, dx$ evaluates to $\frac{3}{4} + 2 \ln 2$. Substituting this back into the volume expression: [ V = \frac{\sqrt{3}}{4} \left( \frac{3}{4} + 2 \ln 2 \right) = \frac{\sqrt{3}}{4} \left( \frac{3 + 8 \ln 2}{4} \right) = \frac{\sqrt{3}(3 + 8 \ln 2)}{16} ]

Comparing this to the form $\frac{\sqrt{3}(p + q\ln 2)}{r}$, we have: $p = 3$ $q = 8$ $r = 16$

We need to find $p + q + r$: [ p + q + r = 3 + 8 + 16 = 27 ]

\boxed{27}