Compute $\int \big( (4z)^5 + 4(4z)^2 \big ) \big( (4z)^3 +1 \big)^{12} dz$ by substitution

Question

The problem is $\int \big( (4z)^5 + 4(4z)^2 \big ) \big( (4z)^3 +1 \big)^{12} dz$. I know a substitution has to be used, but having the $4$ in front of the $4z$ is confusing me and I don't know how to get rid of it.

Answer 1

\begin{align} &(4z \mapsto y)\\ &\int \left((4z)^5+4(4z)^2\right)\left((4z)^3+1\right)^{12}dz\\ &=\frac14\int y^2\left(y^3+4\right)\left(y^3+1\right)^{12}dy\\ &\\ &(y^3\mapsto x)\\ &=\frac1{12}\int (x+4)(x+1)^{12}dx\\ &=\frac1{12}\int (x+1)(x+1)^{12}dx+\frac1{12}\int 3(x+1)^{12}dx\\ &=\frac1{12\cdot14}(x+1)^{14}+\frac1{4\cdot13}(x+1)^{13}\\ &=\frac1{12\cdot14}(\sqrt[3]{4z}+1)^{14}+\frac1{4\cdot13}(\sqrt[3]{4z}+1)^{13}\\ \end{align}