A questions about integration

Question

Let $c,m, a$ be positive integer numbers. I have the following integral: $$2\int_{0}^{\infty}\frac{(x^2+c)^{\frac{m}{2}}}{x}\mathrm{e}^{-ax}\mathrm{d}x.$$ Although, I tried to solve it by Maple, I could not get the answer

Answer 1

As gammatester already commented, there is a major problem around $x=0$. Developed as a Taylor series around $x=0$, we have $$ \left(x^2+c\right)^{m/2}e^{-a x}=c^{m/2}-a c^{m/2}x+\frac{1}{2} c^{\frac{m}{2}-1} \left(a^2 c+m\right)x^2+O\left(x^3\right)$$ which makes $$\frac{(x^2+c)^{m/2}}{x}\mathrm{e}^{-ax}=\frac{c^{m/2}}{x}-a c^{m/2}+\frac{1}{2} c^{\frac{m}{2}-1} \left(a^2 c+m\right)x+O\left(x^2\right)$$

Any $c\neq 0$ would make the integral not converging for a lower bound equal to $0$.