Indefinite integration of $\frac{\exp(x)}{\sqrt(x)}$

Question

How to integrate $\frac{\exp(x)}{\sqrt(x)}$ with respect to $x$?

Answer 1

Let $x=t^{2}$ \begin{align} \int \frac{\mathrm{e}^{x}}{\sqrt{x}} dx &= 2 \int \mathrm{e}^{t^{2}} dt \\ &= \sqrt{\pi} \mathrm{erfi}(t) \\ &= \sqrt{\pi} \mathrm{erfi}(\sqrt{x}) + C \end{align} where $$\mathrm{erfi}(z) = \frac{2}{\sqrt{\pi}} \int\limits_{0}^{z} \mathrm{e}^{t^{2}} dt$$ is the imaginary error function.

Answer 2

Alternatively, hypergeometric: $$ \int \frac{e^x}{\sqrt{x}}\;dx = 2\,\sqrt {x}\;{\mbox{$_1$F$_1$}\left(\frac{1}{2};\frac{3}{2};\,x\right)} + C $$