For what values of $k$ does the integral $\int_{0^+}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}$ converge? For what values of $k$ does the integral $$\int_{0^+}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}$$ converge?
I believe it does not converge for any $k$. 
 A: $\int_{0^+}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}=\int_{0^+}^{1} \frac{dx}{x^k \sqrt{x+x^2}}+\int_{1}^{2} \frac{dx}{x^k \sqrt{x+x^2}}+\int_{2}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}$
If $0<x<1$,  $\sqrt{x+x^2}\ge \sqrt{x}$ hence $\int_{0^+}^{1} \frac{dx}{x^k \sqrt{x+x^2}}\le \int_{0^+}^{1} x^{-k-\frac{1}{2}} dx<\infty$ if $-k+\frac{1}{2}>0$, that is $k<\frac{1}{2}$.
If $x>2$,  $\sqrt{x+x^2}\ge x$ hence $\int_{2}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}\le \int_{2}^{\infty} x^{-k-1} dx<\infty$ if $-k<0$, that is $k>0$.
So it converges for $0<k<\frac{1}{2}$.
If $k\ge \frac{1}{2}$,$\int_{0^+}^{1} \frac{dx}{x^k \sqrt{x+x^2}}=\infty$ while if $k\le 0$, $\int_{2}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}=\infty$.
A: "Obviously", the first problem is around $x=0$.
Let us look at the denominator and its Taylor expansion $$x^k\sqrt{x+x^2}=x^k\Big(\sqrt{x}+\frac{x^{3/2}}{2}+O\left(x^{5/2}\right)\Big)\approx x^{k+\frac 12}+\cdots$$ So, close to $x=0$ we have $$\int x^{-(k+\frac 12)}\,dx=\frac{x^{\frac{1}{2}-k}}{\frac{1}{2}-k}$$ from which I am sure that you can conclude that $k<\frac 12$ is required.
The second problem arise for infinitely large values of $x$. Using the same approach$$x^k\sqrt{x+x^2}=x^k\Big(x+\frac{1}{2}-\frac{1}{8 x}+O\left(\frac{1}{x^2}\right)\Big)\approx x^{k+1}+\cdots$$ So, close to $x=\infty$ we have $$\int x^{-(k+1)}\,dx=-\frac{x^{-k}}{k}$$ which requires $k>0$. 
Just for illustration purposes, I give below the result of the numerical integration for a few values of $k$; they clearly show the vertical asymptotes and the $\large U$ shape of the function $f(k)=\int_{0^+}^{\infty} \frac{dx}{x^k \sqrt{x+x^2}}$
$$\left(
\begin{array}{cc}
 0.005 & 201.408 \\
 0.010 & 101.430 \\  
 0.050 & 21.6196 \\
 0.100 & 11.9058 \\
 0.200 & 7.74848 \\
 0.300 & 7.74848 \\
 0.400 & 11.9058 \\
 0.450 & 21.6196 \\
 0.490 & 101.430 \\
 0.495 & 201.408
\end{array}
\right)$$
