How to evaluate $\int \frac{dx}{(2x+1)\sqrt{3x+2}}$

Question

Evaluate $$\int \frac{dx}{(2x+1)\sqrt{3x+2}}$$

I used the substitution,$$t=3x+2$$

Which leads to $$dt=3dx$$

But then the denominator becomes much more complex to simplify(I can show my working if necessary). Is my substitution wrong? Please Help!

Answer 1

If you do $x=\dfrac{y^2-2}3$ and $\mathrm dx=\dfrac23y\,\mathrm dy$, then your function becomes a rational function (because then $3x+2=y^2$).

Answer 2

You should substitute $3x+2 = t^2$, then the given integral can be solved in the next step using the direct formula.

Answer 3

Hint:

Set $\sqrt{3x+2}=y\implies\dfrac{3dx}{2\sqrt{3x+2}}=dy$

$3x+2=y^2\iff2x+1=?$