Answer
$\ln 5-\ln 4+2\ln x+\dfrac {1}{3}\ln \left( 1-x\right) -2\ln \left( x+1\right) $
Work Step by Step
We use the rules of logarithms to obtain:
$\ln \dfrac {5x^{2}\sqrt [3] {1-x}}{4\left( x+1\right) ^{2}}=\ln 5+\ln x^{2}+\ln \left( 1-x\right) ^{\dfrac {1}{3}}-\ln 4-\ln \left( x+1\right) ^{2}=\ln 5-\ln 4+2\ln x+\dfrac {1}{3}\ln \left( 1-x\right) -2\ln \left( x+1\right) $