Optimal. Leaf size=20 \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2} \]
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Rubi [A] time = 0.0016797, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {74} \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 74
Rubi steps
\begin{align*} \int \sqrt{1-x} x \sqrt{1+x} \, dx &=-\frac{1}{3} (1-x)^{3/2} (1+x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0025343, size = 15, normalized size = 0.75 \[ -\frac{1}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{3}{2}}} \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 = 1.65474, size = 15, normalized size = 0.75 \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53953, size = 54, normalized size = 2.7 \begin{align*} \frac{1}{3} \,{\left (x^{2} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 48.2723, size = 95, normalized size = 4.75 \begin{align*} - 2 \left (\begin{cases} \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} - \frac{\left (1 - x\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37071, size = 23, normalized size = 1.15 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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