int$\_(1)^(0)2$pi$($t$^(2)+$t$)$sqrt$((-3$t$^(2))^(2)+(2$t$+1)^(2))$dt

$$

${\displaystyle \underset{0}{\overset{1}{}}}2(t^2+t)\sqrt[\phantom{2}]{(-3t^2{)}^2+(2t+1{)}^2}dt$

${\displaystyle \underset{1}{\overset{0}{}}}2(1-t^3)\sqrt[\phantom{2}]{(-3t^2{)}^2+(2t+1{)}^2}dt$

${\displaystyle \underset{0}{\overset{1}{}}}2(1-t^3)\sqrt[\phantom{2}]{(-3t^2{)}^2+(2t+1{)}^2}dt$