Closed form solution to $\int_0^{\infty} x(1+({\frac{x}{\lambda}})^{(-\alpha-1)}) dx$

Question

Is there a closed form solution for this integral?

$${\int_0^{\infty} x(1+{\frac{x}{\lambda}})^{(-\alpha-1)} dx}$$

where $$\alpha>1$$ $$\lambda>0$$

It's in Pareto statistical distribution family and I'm trying to find an analytical solution.

Answer 1

The integral in the denominator looks like the expected value of a Lomax distribution. (some constants missing) I read that you need $\alpha >1$, otherwise undefined. Looking at the Wikipedia material, It appears the solution is

$$\int_0^{\infty} x\left(1+{\frac{x}{\lambda}}\right)^{(-\alpha-1)}dx = \frac{\lambda^2}{\alpha(\alpha -1)}$$

but verify.

Answer 2

$$I=\int_{0}^{\infty} \frac{x}{(1+x/b)^{(1+a)}} dx =b^{(1+a)} \int_{0}^{\infty} \left( \frac{1}{(b+x)^a} - \frac{b}{(b+x)^{1+a}} \right)=b^{1+a} \left( \frac{(b+x)^{1-a}}{1-a}+b \frac{(b+x)^{-a}}{a} \right)_{0}^{\infty}=\frac{b^2}{a(a-1)},~ a > 1$$