Evaluate: integral of (1+ln(x))^2 x

Expression: $\int{ {\left( 1+\ln\left({x}\right) \right)}^{2} } \mathrm{d} x$

Use ${\left( a+b \right)}^{2}={a}^{2}+2ab+{b}^{2}$ to expand the expression

$\int{ 1+2\ln\left({x}\right)+{\ln\left({x}\right)}^{2} } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\int{ 1 } \mathrm{d} x+\int{ 2\ln\left({x}\right) } \mathrm{d} x+\int{ {\ln\left({x}\right)}^{2} } \mathrm{d} x$

Use $\int{ 1 } \mathrm{d} x=x$ to evaluate the integral

$x+\int{ 2\ln\left({x}\right) } \mathrm{d} x+\int{ {\ln\left({x}\right)}^{2} } \mathrm{d} x$

Evaluate the indefinite integral

$x+2x \times \ln\left({x}\right)-2x+\int{ {\ln\left({x}\right)}^{2} } \mathrm{d} x$

Evaluate the indefinite integral

$x+2x \times \ln\left({x}\right)-2x+{\ln\left({x}\right)}^{2} \times x-2x \times \ln\left({x}\right)+2x$

Simplify the expression

$x+{\ln\left({x}\right)}^{2} \times x$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }x+{\ln\left({x}\right)}^{2} \times x+C,& C \in ℝ\end{array}$