Definite integration of a high order exponential function mixed with rational function

Question

I would like to solve the integral

$$\int_{x>0}xe^{ax^m+bx^n}~dx,\qquad m>n>0$$

Answer 1

$\int_0^\infty xe^{ax^m+bx^n}~dx$

$=\int_0^\infty x^\frac{1}{m}~e^{ax}e^{bx^\frac{n}{m}}~d\left(x^\frac{1}{m}\right)$

$=\dfrac{1}{m}\int_0^\infty x^{\frac{2}{m}-1}e^{ax}e^{bx^\frac{n}{m}}~dx$

$=\int_0^\infty\sum\limits_{k=0}^\infty\dfrac{b^kx^{\frac{nk+2}{m}-1}e^{ax}}{mk!}dx$

$=\sum\limits_{k=0}^\infty\dfrac{b^k\Gamma\left(\dfrac{nk+2}{m}\right)}{(-a)^\frac{nk+2}{m}~mk!}$

$=\dfrac{1}{(-a)^\frac{2}{m}m}~_1\Psi_0\left[\begin{matrix}\left(\dfrac{2}{m},\dfrac{n}{m}\right)\\-\end{matrix};\dfrac{b}{(-a)^\frac{n}{m}}\right]$ (according to http://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function)