calculus - Evaluating $int_0^infty dn , frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate,

$$\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$$

for $x\in\mathbb{R}$. Using Mathematica I have found the integral to be,

$$\frac{1}{3x^{2/3}} \left[ x^{1/3}\Gamma(0,-\log(x)/x)-\Gamma(0,-2\log(x)/x)\right]$$

providing $x<1$. So I just need to express the integral in terms of incomplete gamma functions, but I haven't found a substitution. Can someone offer a hint (and not a complete solution)?

Answer

Hint. Recall that the incomplete gamma function may be defined as $$ \Gamma(s,a)=\int_a^{\infty} t^{s-1}e^{-t}dt,\quad a>0, s \in \mathbb{R}. \tag1 $$ Observe that, for $0