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 = 0.0044375, 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} \[ \frac{2}{45} (3 x+1)^{5/2}-\frac{2}{27} (3 x+1)^{3/2} \]
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*}
Mathematica [A] time = 0.005936, 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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Maple [A] time = 0.002, size = 15, normalized size = 0.6 \begin{align*}{\frac{18\,x-4}{135} \left ( 1+3\,x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949593, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{45} \,{\left (3 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (3 \, 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 = 0.412359, size = 54, normalized size = 2. \begin{align*} \frac{2}{135} \,{\left (27 \, x^{2} + 3 \, x - 2\right )} \sqrt{3 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.854789, size = 39, normalized size = 1.44 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09895, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{45} \,{\left (3 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{27} \,{\left (3 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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