My problem is to simplify the following integral:

$\displaystyle

G=\int_{x=0}^{1}{exp{(tx+kx^{1.5})}}\,dx

$

I have a solution, but can't understand it without explanations. The solution is following:

1) $\displaystyle k>{\frac{2}{3}t}$

$\displaystyle G_1={\frac{4}{3}}{\frac{\sqrt{\pi}}{k}{t^{1.5}}}ex p({{\frac{4}{27}}{t^3}k^{-2})$

2) $\displaystyle k<{\frac{2}{3}t}$

$\displaystyle G_2={\frac{1}{2}}{{\sqrt{\frac{\pi}{a}}}exp{\left( {\frac{b^2-ac}{c}}\right)erfc\left({\frac{b}{\sqrt{a}}}\right )$, where $\displaystyle a={\frac{3}{8}}t$, $\displaystyle b={\frac{1}{2}}t-{\frac{3}{4}}k$, $\displaystyle c=k-t$

The second case is clear: we expand the exponential term to the 2nd order Taylor series around u=1, simplify the expression, change variable, set the lower integration linit from 0 to $\displaystyle -\infty$ and get the complimentary error function. But the first case result is not clear at all. Please, help me to understand. Thank you.