Optimal. Leaf size=27 \[ \frac {2}{45} (3 x+1)^{5/2}-\frac {2}{27} (3 x+1)^{3/2} \]
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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 = {43} \[ \frac {2}{45} (3 x+1)^{5/2}-\frac {2}{27} (3 x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
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 \[ \frac {2}{135} (3 x+1)^{3/2} (9 x-2) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 19, normalized size = 0.70 \[ \frac {2}{135} \, {\left (27 \, x^{2} + 3 \, x - 2\right )} \sqrt {3 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 19, normalized size = 0.70 \[ \frac {2}{45} \, {\left (3 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (3 \, x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.56 \[ \frac {2 \left (3 x +1\right )^{\frac {3}{2}} \left (9 x -2\right )}{135} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 19, normalized size = 0.70 \[ \frac {2}{45} \, {\left (3 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (3 \, x + 1\right )}^{\frac {3}{2}} \]
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 \[ \frac {2\,{\left (3\,x+1\right )}^{3/2}\,\left (9\,x-2\right )}{135} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 39, normalized size = 1.44 \[ \frac {2 x^{2} \sqrt {3 x + 1}}{5} + \frac {2 x \sqrt {3 x + 1}}{45} - \frac {4 \sqrt {3 x + 1}}{135} \]
Verification of antiderivative is not currently implemented for this CAS.
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