TITLE: Evaluate $\int _{ 0 }^{ 1 }{ \left( { x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 2 } \right) \sqrt { 4{ x }^{ 3 }+5{ x }^{ 2 }+10 } \; dx } $
QUESTION [11 upvotes]: Evaluate
   $$\int _{ 0 }^{ 1 }{ \left( { x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 2
 } \right) \sqrt { 4{ x }^{ 3 }+5{ x }^{ 2 }+10 } \; dx } $$

The question look's like there is a nice method to do it, but I can't figure out. Can someone provide some hint or answer?.

REPLY [17 votes]: $$\begin{align}
\int_0^1(x^5+x^4+x^2)\sqrt{4x^3+5x^2+10}dx &=\int_0^1(x^4+x^3+x)\sqrt{4x^5+5x^4+10x^2}dx\cr
&={1\over20}\int_0^{19}\sqrt{u}du={19^{3/2}\over30}\cr
\end{align}$$