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 = 0.0039135, antiderivative size = 23, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0031991, size = 16, normalized size = 0.7 \[ \frac{2}{15} (x+1)^{3/2} (3 x-2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 13, normalized size = 0.6 \begin{align*}{\frac{-4+6\,x}{15} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926539, size = 20, normalized size = 0.87 \begin{align*} \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (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 = 1.79048, size = 46, normalized size = 2. \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + x - 2\right )} \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.839955, size = 34, normalized size = 1.48 \begin{align*} \frac{2 x^{2} \sqrt{x + 1}}{5} + \frac{2 x \sqrt{x + 1}}{15} - \frac{4 \sqrt{x + 1}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09204, size = 20, normalized size = 0.87 \begin{align*} \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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