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To find the general form of the antiderivative (indefinite integral) of the function \( f(x) = 2x^3 + 2x \), you integrate each term separately:

1. The integral of \( 2x^3 \) is: \[ \int 2x^3 \, dx = \frac{2}{4}x^4 = \frac{1}{2}x^4 \]

2. The integral of \( 2x \) is: \[ \int 2x \, dx = x^2 \]

Combining these, the general antiderivative is: \[ F(x) = \frac{1}{2}x^4 + x^2 + C \]

where \( C \) is the constant of integration.