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Let $u = 1 + 2x$. Then, $du = 2dx$, so $dx = \frac{1}{2}du$. Also, $x = \frac{u-1}{2}$.
Substituting these into the integral, we get: $\int x\sqrt{1+2x}dx = \int \frac{u-1}{2}\sqrt{u}\frac{1}{2}du = \frac{1}{4}\int (u-1)\sqrt{u}du = \frac{1}{4}\int (u^{3/2} - u^{1/2})du$
Now, we integrate term by term: $\frac{1}{4}\int (u^{3/2} - u^{1/2})du = \frac{1}{4} \left( \int u^{3/2}du - \int u^{1/2}du \right) = \frac{1}{4} \left( \frac{u^{5/2}}{5/2} - \frac{u^{3/2}}{3/2} \right) + C = \frac{1}{4} \left( \frac{2}{5}u^{5/2} - \frac{2}{3}u^{3/2} \right) + C = \frac{1}{10}u^{5/2} - \frac{1}{6}u^{3/2} + C$
Substitute $u = 1 + 2x$ back into the expression: $\frac{1}{10}(1+2x)^{5/2} - \frac{1}{6}(1+2x)^{3/2} + C$
We can simplify this expression further by factoring out $(1+2x)^{3/2}$: $(1+2x)^{3/2} \left( \frac{1}{10}(1+2x) - \frac{1}{6} \right) + C = (1+2x)^{3/2} \left( \frac{3(1+2x) - 5}{30} \right) + C = (1+2x)^{3/2} \left( \frac{3+6x-5}{30} \right) + C = (1+2x)^{3/2} \left( \frac{6x-2}{30} \right) + C = \frac{1}{15}(3x-1)(1+2x)^{3/2} + C$
Final Answer: $\frac{1}{10}(1+2x)^{5/2} - \frac{1}{6}(1+2x)^{3/2} + C$ or $\frac{1}{15}(3x-1)(1+2x)^{3/2} + C$
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