How to proceed $\int \frac{dx}{(x-2) \left(1+\sqrt{7 x-10-x^2}\right)} $ 
$$I=\int \frac{dx}{(x-2) \left(1+\sqrt{7 x-10-x^2}\right)} $$

I have taken the second term of denominator (in square root) as $(x - 2)t$. But cannot go further. Please suggest how to proceed or any alternate method to solve it.

Note that $7x-10-x^2 = \frac94-(\frac72-x)^2$. So, substitute  $\frac72-x=\frac32\sin t$ to integrate as follows
\begin{align}
& \int \frac{dx}{(x-2)(1+\sqrt{7x-10-x^2})}\\
= & - \int \frac{2\cos t}{(1-\sin t)(2+3\cos t)}dt\\
= & - \int \left ( \frac{\cos t}{1-\sin t}-\frac {3\sin t}{2+3\cos t} - \frac {3}{2+3\cos t} \right)dt\\
=& \>\ln(1-\sin t) -\ln(2+3\cos t)+\frac3{\sqrt5}\tanh^{-1}\frac{\tan\frac t2}{\sqrt5}+C
\end{align}
