Optimal. Leaf size=27 \[ \frac{1}{10} (2 x+1)^{5/2}-\frac{1}{6} (2 x+1)^{3/2} \]
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Rubi [A] time = 0.0048999, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{1}{10} (2 x+1)^{5/2}-\frac{1}{6} (2 x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x \sqrt{1+2 x} \, dx &=\int \left (-\frac{1}{2} \sqrt{1+2 x}+\frac{1}{2} (1+2 x)^{3/2}\right ) \, dx\\ &=-\frac{1}{6} (1+2 x)^{3/2}+\frac{1}{10} (1+2 x)^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0057128, size = 18, normalized size = 0.67 \[ \frac{1}{15} (2 x+1)^{3/2} (3 x-1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.6 \begin{align*}{\frac{3\,x-1}{15} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924912, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{10} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1}{6} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09458, size = 49, normalized size = 1.81 \begin{align*} \frac{1}{15} \,{\left (6 \, x^{2} + x - 1\right )} \sqrt{2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.844147, size = 36, normalized size = 1.33 \begin{align*} \frac{2 x^{2} \sqrt{2 x + 1}}{5} + \frac{x \sqrt{2 x + 1}}{15} - \frac{\sqrt{2 x + 1}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05702, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{10} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1}{6} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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