1.68: Curve from second derivative and normal equation

A curve is such that $\dfrac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} = (2x + 3)^{-\frac{1}{2}}$. The curve has a gradient of $5$ at the point where $x = 3$ and passes through the point $\left(\dfrac{1}{2}, -\dfrac{1}{3}\right)$. Find the equation of the curve.

Find the equation of the normal to the curve at the point where $x = 3$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.