1.86: Partial fractions and shaded area

Show that $\dfrac{1}{x + 1} + \dfrac{2}{3x + 10}$ can be written as $\dfrac{5x + 12}{3x^{2} + 13x + 10}$.

The diagram shows part of the curve $y = \dfrac{5x + 12}{3x^{2} + 13x + 10}$, the line $x = 2$ and a straight line of gradient $1$. The curve intersects the $y$-axis at the point $P$. The line of gradient $1$ passes through $P$ and intersects the $x$-axis at the point $Q$. Find the area of the shaded region, giving your answer in the form $a + \dfrac{2}{3}\ln(b\sqrt{3})$, where $a$ and $b$ are constants.