$\int \frac{6 x+7}{\sqrt{(x-5)(x-4)}} d x=\int \frac{(6 x+7) d x}{\sqrt{x^2-9 x+20}}$
Let $6 x+7=\mathrm{A} \times \frac{d}{d x}\left(x^2-9 x+20\right)+\mathrm{B}$
$\Rightarrow 6 x+7=\mathrm{A}(2 x-9)+\mathrm{B}$
$\Rightarrow 2 \mathrm{~A}=6 \Rightarrow \mathrm{A}=3 \& 7=-9 \mathrm{~A}+\mathrm{B} \Rightarrow \mathrm{B}=34$
$\therefore \mathrm{I}=3 \int \frac{2 x-9}{\sqrt{x^2-9 x+20}} d x+34 \int \frac{d x}{\sqrt{x^2-9 x+20}}$
Let $\mathrm{I}=3 \mathrm{I}_1+34 \mathrm{I}_2+\mathrm{C}$
$\therefore \mathrm{I}_1=\int \frac{d t}{\sqrt{t}}=2 t^{1 / 2}=2 \sqrt{x^2-9 x+20}$
$\mathrm{I}_2=\int \frac{d x}{\sqrt{\left(x-\frac{9}{2}\right)^2-\left(\frac{1}{2}\right)^2}}$
$=\log \left|x-\frac{9}{2}+\sqrt{\left(x-\frac{9}{2}\right)^2-\left(\frac{1}{2}\right)^2}\right|$
$\therefore I=6 \sqrt{x^2-9 x+20}+34 \log \left|\left(x-\frac{9}{2}\right)+\sqrt{x^2-9 x+20}\right|+C$