1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_1^3 (1 + 8xy) \, dy \, dx.$$\n\n2. **Recall the formula for double integrals:** The integral over a rectangular region is computed by integrating the inner integral first, then the outer integral:\n$$\int_a^b \int_c^d f(x,y) \, dy \, dx = \int_a^b \left( \int_c^d f(x,y) \, dy \right) dx.$$\n\n3. **Integrate with respect to $y$ first:** Fix $x$ and integrate $$1 + 8xy$$ from $y=1$ to $y=3$.\n\n$$\int_1^3 (1 + 8xy) \, dy = \int_1^3 1 \, dy + \int_1^3 8xy \, dy = (y \big|_1^3) + 8x \left( \frac{y^2}{2} \bigg|_1^3 \right) = (3 - 1) + 8x \left( \frac{9 - 1}{2} \right) = 2 + 8x \times 4 = 2 + 32x.$$\n\n4. **Now integrate with respect to $x$ from 0 to 1:**\n\n$$\int_0^1 (2 + 32x) \, dx = \int_0^1 2 \, dx + \int_0^1 32x \, dx = 2x \big|_0^1 + 16x^2 \big|_0^1 = 2 + 16 = 18.$$\n\n5. **Final answer:** The value of the double integral is $$\boxed{18}.$$