5. Evaluate the definite integral: $\int_{0}^{3} 16 x\left(x^{2}+1\right)^{3} d x$

Step 1 ：Define the variable x.

Step 2 ：Define the function as \(f = 16x(x^2 + 1)^3\).

Step 3 ：Compute the definite integral of the function from 0 to 3.

Step 4 ：The result of the definite integral from 0 to 3 of the function \(16x(x^2+1)^3\) is 19998.

Step 5 ：The final answer is \(\boxed{19998}\).