[THOUGHT] The volume of the solid is given by integrating the area of equilateral triangles perpendicular to the x-axis over the interval [1, 2]. The side length of each triangle is the y-value of the curve, ( s(x) = \sqrt{x + x \ln x} ). The area of an equilateral triangle is ( A(x) = \frac{\sqrt{3}}{4} s(x)^2 = \frac{\sqrt{3}}{4} (x + x \ln x) ). Thus, the volume is: [ V = \frac{\sqrt{3}}{4} \int_1^2 (x + x \ln x) , dx. ] We compute the integral ( I = \int_1^2 (x + x \ln x) , dx ) using symbolic integration in Python. The result will be expressed in terms of ( \ln 2 ), and we will match it to the form ( \frac{\sqrt{3}(p + q \ln 2)}{r} ) to find ( p, q, r ). [/THOUGHT] [PYTHON] from sympy import symbols, integrate, sqrt, ln, simplify

x = symbols('x')

Define the integrand

integrand = x + x * ln(x)

Compute the definite integral from 1 to 2

I = integrate(integrand, (x, 1, 2))

Simplify the result

I_simplified = simplify(I) I_simplified [/PYTHON]

[THOUGHT] The Python output gives the integral result as ( \frac{3}{4} + \ln(4) ). Rewriting ( \ln(4) = 2 \ln 2 ), the integral becomes ( \frac{3}{4} + 2 \ln 2 ). The volume is ( \frac{\sqrt{3}}{4} ) times this, leading to ( \frac{\sqrt{3}(3 + 8 \ln 2)}{16} ). Thus, ( p = 3 ), ( q = 8 ), ( r = 16 ), and ( p + q + r = 27 ). [/THOUGHT] \boxed{27}