Optimal. Leaf size=20 \[ -\frac {1}{3} (1-x)^{3/2} (1+x)^{3/2} \]
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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 = {75}
\begin {gather*} -\frac {1}{3} (1-x)^{3/2} (x+1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
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 \begin {gather*} -\frac {1}{3} \left (1-x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 1.00
method | result | size |
gosper | \(-\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{3}\) | \(15\) |
default | \(\frac {\sqrt {1-x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\) | \(20\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \sqrt {1+x}\, \left (x^{2}-1\right ) \left (-1+x \right )}{3 \sqrt {1-x}\, \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 11, normalized size = 0.55 \begin {gather*} -\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 19, normalized size = 0.95 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (15) = 30\).
time = 21.53, size = 129, normalized size = 6.45 \begin {gather*} - 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}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \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}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (14) = 28\).
time = 1.57, size = 43, normalized size = 2.15 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \frac {\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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