To determine the antiderivative (integral) of $f(x) = (2x^3 + 3x)^7 (2x^2 + 1)$, we can apply substitution.

Let: $u = 2x^3 + 3x$ Then, the derivative $\frac{du}{dx} = 6x^2 + 3$, or equivalently, $du = (6x^2 + 3) dx$.

This integral setup leads us to look for the solution among the provided options.

Evaluating each option, the correct choice is:

c. $18(2x^3 + 3x)^8 + C$

Would you like more detailed steps?

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Here are five related questions you might find interesting:

1. What is the general method to solve integrals with power functions and linear terms?
2. How does substitution help simplify complex integrals?
3. What are some other examples of integrals that use similar techniques?
4. Can this integral be evaluated with different substitution methods?
5. How can we verify the solution by differentiating the answer?

Tip: Always check your solution by differentiating to ensure it matches the original function.