Optimal. Leaf size=48 \[ \frac{1}{2} \sqrt{1-x^2} x^3+\frac{2 x^3}{3}-\frac{1}{4} \sqrt{1-x^2} x+\frac{1}{4} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0895353, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6742, 279, 321, 216} \[ \frac{1}{2} \sqrt{1-x^2} x^3+\frac{2 x^3}{3}-\frac{1}{4} \sqrt{1-x^2} x+\frac{1}{4} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 279
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x^2 \left (\sqrt{1-x}+\sqrt{1+x}\right )^2 \, dx &=\int \left (2 x^2+2 x^2 \sqrt{1-x^2}\right ) \, dx\\ &=\frac{2 x^3}{3}+2 \int x^2 \sqrt{1-x^2} \, dx\\ &=\frac{2 x^3}{3}+\frac{1}{2} x^3 \sqrt{1-x^2}+\frac{1}{2} \int \frac{x^2}{\sqrt{1-x^2}} \, dx\\ &=\frac{2 x^3}{3}-\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{2} x^3 \sqrt{1-x^2}+\frac{1}{4} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{2 x^3}{3}-\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{2} x^3 \sqrt{1-x^2}+\frac{1}{4} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0431275, size = 42, normalized size = 0.88 \[ \frac{1}{12} \left (\left (6 \sqrt{1-x^2}+8\right ) x^3-3 \sqrt{1-x^2} x+3 \sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 59, normalized size = 1.2 \begin{align*}{\frac{2\,{x}^{3}}{3}}+{\frac{1}{4}\sqrt{1-x}\sqrt{1+x} \left ( 2\,{x}^{3}\sqrt{-{x}^{2}+1}-x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6985, size = 46, normalized size = 0.96 \begin{align*} \frac{2}{3} \, x^{3} - \frac{1}{2} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{1}{4} \, \sqrt{-x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22624, size = 134, normalized size = 2.79 \begin{align*} \frac{2}{3} \, x^{3} + \frac{1}{4} \,{\left (2 \, x^{3} - x\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{2} \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11715, size = 84, normalized size = 1.75 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{3} - 2 \,{\left (x + 1\right )}^{2} + \frac{1}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2 \, x + \frac{1}{2} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) + 2 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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