Optimal. Leaf size=27 \[ -\frac {1}{6} (1+2 x)^{3/2}+\frac {1}{10} (1+2 x)^{5/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {45}
\begin {gather*} \frac {1}{10} (2 x+1)^{5/2}-\frac {1}{6} (2 x+1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {1}{15} (1+2 x)^{3/2} (-1+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 20, normalized size = 0.74
method | result | size |
gosper | \(\frac {\left (1+2 x \right )^{\frac {3}{2}} \left (3 x -1\right )}{15}\) | \(15\) |
risch | \(\frac {\left (6 x^{2}+x -1\right ) \sqrt {1+2 x}}{15}\) | \(18\) |
trager | \(\left (\frac {2}{5} x^{2}+\frac {1}{15} x -\frac {1}{15}\right ) \sqrt {1+2 x}\) | \(19\) |
derivativedivides | \(-\frac {\left (1+2 x \right )^{\frac {3}{2}}}{6}+\frac {\left (1+2 x \right )^{\frac {5}{2}}}{10}\) | \(20\) |
default | \(-\frac {\left (1+2 x \right )^{\frac {3}{2}}}{6}+\frac {\left (1+2 x \right )^{\frac {5}{2}}}{10}\) | \(20\) |
meijerg | \(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+2 x \right )^{\frac {3}{2}} \left (-6 x +2\right )}{15}}{8 \sqrt {\pi }}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.40, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{15} \, {\left (6 \, x^{2} + x - 1\right )} \sqrt {2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 36, normalized size = 1.33 \begin {gather*} \frac {2 x^{2} \sqrt {2 x + 1}}{5} + \frac {x \sqrt {2 x + 1}}{15} - \frac {\sqrt {2 x + 1}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 14, normalized size = 0.52 \begin {gather*} \frac {{\left (2\,x+1\right )}^{3/2}\,\left (6\,x-2\right )}{30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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