Optimal. Leaf size=20 \[ -\frac {1}{3} (1-x)^{3/2} (x+1)^{3/2} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 0.75 \[ -\frac {1}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 19, normalized size = 0.95 \[ \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.40, size = 43, normalized size = 2.15 \[ \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.75 \[ -\frac {\left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 11, normalized size = 0.55 \[ -\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 19, normalized size = 0.95 \[ \frac {\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 35.69, size = 95, normalized size = 4.75 \[ - 2 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 2 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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