Optimal. Leaf size=23 \[ \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 261} \[ \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 261
Rubi steps
\begin {align*} \int x \left (1+\sqrt {1-x} \sqrt {1+x}\right ) \, dx &=\int \left (x+x \sqrt {1-x^2}\right ) \, dx\\ &=\frac {x^2}{2}+\int x \sqrt {1-x^2} \, dx\\ &=\frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 25, normalized size = 1.09 \[ \frac {1}{2} \, x^{2} + \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 54, normalized size = 2.35 \[ \frac {1}{2} \, {\left (x + 1\right )}^{2} + \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - x - 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.13 \[ \frac {x^{2}}{2}+\frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (x^{2}-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 17, normalized size = 0.74 \[ \frac {1}{2} \, x^{2} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 35, normalized size = 1.52 \[ \frac {x^2}{2}-\frac {\sqrt {1-x}\,\left (-\frac {x^3}{3}-\frac {x^2}{3}+\frac {x}{3}+\frac {1}{3}\right )}{\sqrt {x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.60, size = 105, normalized size = 4.57 \[ - x + \frac {\left (x + 1\right )^{2}}{2} - 2 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 2 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) - 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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