Optimal. Leaf size=19 \[ x^2-\frac{2}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A] time = 0.0541514, antiderivative size = 19, normalized size of antiderivative = 1., 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 = {6742, 261} \[ x^2-\frac{2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 261
Rubi steps
\begin{align*} \int x \left (\sqrt{1-x}+\sqrt{1+x}\right )^2 \, dx &=\int \left (2 x+2 x \sqrt{1-x^2}\right ) \, dx\\ &=x^2+2 \int x \sqrt{1-x^2} \, dx\\ &=x^2-\frac{2}{3} \left (1-x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0206148, size = 19, normalized size = 1. \[ x^2-\frac{2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 1.3 \begin{align*}{x}^{2}+{\frac{2\,{x}^{2}-2}{3}\sqrt{1-x}\sqrt{1+x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60683, size = 20, normalized size = 1.05 \begin{align*} x^{2} - \frac{2}{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 = 1.25989, size = 62, normalized size = 3.26 \begin{align*} x^{2} + \frac{2}{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 [A] time = 79.1557, size = 110, normalized size = 5.79 \begin{align*} - \frac{x^{3}}{3} - x + \frac{\left (x + 1\right )^{3}}{3} - 4 \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 ) + 4 \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11003, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} +{\left (x + 1\right )}^{2} - 2 \, x - 2 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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