Optimal. Leaf size=27 \[ -\frac {2}{27} (1+3 x)^{3/2}+\frac {2}{45} (1+3 x)^{5/2} \]
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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 {2}{45} (3 x+1)^{5/2}-\frac {2}{27} (3 x+1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x \sqrt {1+3 x} \, dx &=\int \left (-\frac {1}{3} \sqrt {1+3 x}+\frac {1}{3} (1+3 x)^{3/2}\right ) \, dx\\ &=-\frac {2}{27} (1+3 x)^{3/2}+\frac {2}{45} (1+3 x)^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {2}{135} (1+3 x)^{3/2} (-2+9 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 20, normalized size = 0.74
method | result | size |
gosper | \(\frac {2 \left (1+3 x \right )^{\frac {3}{2}} \left (9 x -2\right )}{135}\) | \(15\) |
trager | \(\left (\frac {2}{5} x^{2}+\frac {2}{45} x -\frac {4}{135}\right ) \sqrt {1+3 x}\) | \(19\) |
derivativedivides | \(-\frac {2 \left (1+3 x \right )^{\frac {3}{2}}}{27}+\frac {2 \left (1+3 x \right )^{\frac {5}{2}}}{45}\) | \(20\) |
default | \(-\frac {2 \left (1+3 x \right )^{\frac {3}{2}}}{27}+\frac {2 \left (1+3 x \right )^{\frac {5}{2}}}{45}\) | \(20\) |
risch | \(\frac {2 \left (27 x^{2}+3 x -2\right ) \sqrt {1+3 x}}{135}\) | \(20\) |
meijerg | \(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+3 x \right )^{\frac {3}{2}} \left (-9 x +2\right )}{15}}{18 \sqrt {\pi }}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.71, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{45} \, {\left (3 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (3 \, 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 = 0.59, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{135} \, {\left (27 \, x^{2} + 3 \, x - 2\right )} \sqrt {3 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 39, normalized size = 1.44 \begin {gather*} \frac {2 x^{2} \sqrt {3 x + 1}}{5} + \frac {2 x \sqrt {3 x + 1}}{45} - \frac {4 \sqrt {3 x + 1}}{135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{45} \, {\left (3 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (3 \, 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 = 0.09, size = 14, normalized size = 0.52 \begin {gather*} \frac {2\,{\left (3\,x+1\right )}^{3/2}\,\left (9\,x-2\right )}{135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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