Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}\,\text{d}x$. How to do this indefinite integral (anti-derivative)?
$$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}\,\text{d}x.$$
I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.

Use Trigonometric substitution,
$$x=\sqrt7\tan\theta$$
Then  $\displaystyle a\sin y+b\cos y=\sqrt{a^2+b^2}\sin\left(y+\arctan \frac ba\right)$ and this
or use  Weierstrass substitution 
