[THOUGHT] I will compute the integral for the volume using Python and simplify it into the required form. [/THOUGHT] [PYTHON] import sympy as sp

x = sp.symbols('x') expr = x + x*sp.log(x) V = sp.sqrt(3)/4 * sp.integrate(expr, (x,1,2)) V_simplified = sp.simplify(V) print("Volume:", V_simplified)

Express V as sqrt(3)(p + qln(2))/r

p, q, r = sp.symbols('p q r', integer=True)

Solve for (p + q ln 2)/r = the coefficient of sqrt(3)

coeff = sp.simplify(V_simplified / sp.sqrt(3)) print("Coefficient without sqrt(3):", coeff) coeff_rational = sp.together(coeff) print("Together:", coeff_rational)

Extract p, q, r manually from coeff

num, den = sp.fraction(coeff_rational) print("Numerator:", num) print("Denominator:", den)

Numerator should be of form p + q*ln(2)

p_extracted = sp.simplify(num.subs(sp.log(2), 0)) q_extracted = sp.simplify(sp.expand((num - p_extracted)/sp.log(2))) r_extracted = den

print("p:", p_extracted) print("q:", q_extracted) print("r:", r_extracted) print("p+q+r:", p_extracted + q_extracted + r_extracted) [/PYTHON]

[THOUGHT] From the Python result, the volume is $\sqrt{3}(3 + \ln 256)/16$. Since $\ln 256 = \ln(2^8) = 8\ln 2$, this becomes $\sqrt{3}(3 + 8\ln 2)/16$. Thus $p=3$, $q=8$, $r=16$, and $p+q+r=27$. $\boxed{27}$