Optimal. Leaf size=23 \[ \frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2} \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \[ \frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x \sqrt {1+x} \, dx &=\int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx\\ &=-\frac {2}{3} (1+x)^{3/2}+\frac {2}{5} (1+x)^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.70 \[ \frac {2}{15} (x+1)^{3/2} (3 x-2) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 15, normalized size = 0.65 \[ \frac {2}{15} \, {\left (3 \, x^{2} + x - 2\right )} \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 15, normalized size = 0.65 \[ \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.57 \[ \frac {2 \left (x +1\right )^{\frac {3}{2}} \left (3 x -2\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 15, normalized size = 0.65 \[ \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 12, normalized size = 0.52 \[ \frac {2\,\left (3\,x-2\right )\,{\left (x+1\right )}^{3/2}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 34, normalized size = 1.48 \[ \frac {2 x^{2} \sqrt {x + 1}}{5} + \frac {2 x \sqrt {x + 1}}{15} - \frac {4 \sqrt {x + 1}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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