Integrating $\int x^{3}e^{x}\sqrt{x\ln{x}}\,\text{d}x$

Question

How do I integrate following integral? I have tried basic substitutions but cannot get through this.

$$\int x^{3}e^{x}\sqrt{x\ln{x}}\,\text{d}x$$

Thanks.

Answer 1

Let $u=\ln x$ ,

Then $x=e^u$

$dx=e^u~du$

$\therefore\int x^3e^x\sqrt{x\ln x}~dx$

$=\int u^\frac{1}{2}e^{\frac{9u}{2}}e^{e^u}~du$

$=\int\sum\limits_{n=0}^\infty\dfrac{u^\frac{1}{2}e^\frac{(2n+9)u}{2}}{n!}~du$

$=\int\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{(2n+9)^ku^{k+\frac{1}{2}}}{2^kn!k!}~du$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{(2n+9)^ku^{k+\frac{3}{2}}}{2^{k—1}n!k!(2k+3)}+C$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{(2n+9)^k(\ln x)^{k+\frac{3}{2}}}{2^{k—1}n!k!(2k+3)}+C$