Answer
$$\frac{1}{{128}}\left( {3 + {e^2}} \right)$$
Work Step by Step
$$\eqalign{
& \int_1^{\sqrt 2 } {{x^3}{{\ln }^3}x} dx \cr
& {\text{Using a computer algebra system }}\left( {{\text{Wolfram website}}} \right){\text{ we obtain}} \cr
& \int_1^{\sqrt 2 } {{x^3}{{\ln }^3}x} dx = \frac{1}{{128}}\left( {3 + {e^2}} \right) \cr} $$
