Finding the CDF of a Sum of I.I.D Continuous Random Variables

Let X$$_1$$ and X$$_2$$ be identically independent distributions(i.i.d) random variables with

$$\Bbb P(X_i \le x) = 1-x^{-1/2}, \quad x \ge 1 \ \text{and} \ i = 1,2$$

Find $$\Bbb P(X_1 + X_2 \le x)$$.

I tried to finding the convolution of $$f_{X_1}$$ and $$f_{X_2}$$ (the density functions for $$X_1$$ and $$X_2$$) and integrating to get the CDF of $$X_1$$ + $$X_2$$ which is what I interpreted the question is asking for. When I integrate it, it got really messy and I could not finish the integration. Am I doing something wrong?

• Wolframalpha gives the answer using hypergeometric functions. I think there is no hope to get simple answer. – NCh Feb 15 at 6:00
• I actually just noticed that the exponent of -1/3 is incorrect. It’s suppose to be -1/2 if that makes a difference. I’ll make the correction now – Niko L Feb 15 at 13:15
• wolframalpha.com/input/… – NCh Feb 15 at 14:36