Solution: Given that the function is \[ f(z) = \frac{z^3}{z^2 + 7z + 12} = \frac{z^3}{(z + 4)(z+3)}. \] The function is analytic everywhere except the points $z = -4,-3$ which are not inside the unit circle. Thus, by the Cauchy's integral theorem, the given integral will be zero, that is, \[ \int_{C} f(z) \ \mathrm{d} z = 0. \]