Re: taking A.D
No the 1/3 is definitely needed...
$\displaystyle \begin{align*} \int{ \left( z^5 + 9z^2 \right) \, \left( z^3 + 1 \right) ^{12}\,\mathrm{d}z} &= \int{ z^2\,\left( z^3 + 9 \right) \, \left( z^3 + 1 \right) ^{12}\,\mathrm{d}z} \\ &= \frac{1}{3} \int{ 3z^2 \,\left( z^3 + 1 + 8 \right) \, \left( z^3 + 1 \right) ^{12}\,\mathrm{d}z } \end{align*}$
So now make the substitution $\displaystyle \begin{align*} u = z^3 + 1 \implies \mathrm{d}u = 3z^2\,\mathrm{d}z \end{align*}$ and the integral becomes
$\displaystyle \begin{align*} \frac{1}{3} \int{ 3z^2 \, \left( z^3 + 1 + 8 \right) \, \left( z^3 + 1 \right) \, \mathrm{d}z } &= \frac{1}{3} \int{ \left( u + 8 \right) \, u^{12} \, \mathrm{d}u } \\ &= \frac{1}{3} \int{ u^{13} + 8u^{12}\,\mathrm{d}u } \\ &= \frac{1}{3} \left( \frac{1}{14} u^{14} + \frac{8}{13} u^{13} \right) + C \\ &= \frac{1}{3} \left[ \frac{1}{14} \left( z^3 + 1 \right) ^{14} + \frac{8}{13} \left( z^3 + 1 \right) ^{13} \right] + C \end{align*}$