Solve the integral of logarithmic functions $\int x\ln\left(x\right)dx$

Find the integral

$\int x^3\log\left(x\right)dx$

Find the integral

$\int\ln\left(\sqrt{x}+\sqrt{1+x}\right)dx$

Find the integral

$\int x^2\ln Xdx$

Find the integral

$\int x^2\:lnx\:dx$

Find the integral

$\int\:x\:ln\:2x\:dx$

Find the integral

$\int xLnxdx$

Find the integral

$\int x\:ln\left(2x\right)dx$