Optimal. Leaf size=48 \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac{1}{2} \sqrt{1-x} x \sqrt{x+1}+\frac{1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0055341, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac{1}{2} \sqrt{1-x} x \sqrt{x+1}+\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 49
Rule 38
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-x} (1+x)^{3/2} \, dx &=-\frac{1}{3} (1-x)^{3/2} (1+x)^{3/2}+\int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{1}{2} \sqrt{1-x} x \sqrt{1+x}-\frac{1}{3} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} x \sqrt{1+x}-\frac{1}{3} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} x \sqrt{1+x}-\frac{1}{3} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0336247, size = 44, normalized size = 0.92 \[ \frac{1}{6} \sqrt{1-x^2} \left (2 x^2+3 x-2\right )-\sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 71, normalized size = 1.5 \begin{align*}{\frac{1}{3}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{1}{6}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52172, size = 38, normalized size = 0.79 \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52655, size = 123, normalized size = 2.56 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} + 3 \, x - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - \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 [B] time = 5.359, size = 165, normalized size = 3.44 \begin{align*} \begin{cases} - i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{x - 1}} - \frac{5 i \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{x - 1}} - \frac{i \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{x - 1}} + \frac{i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{1 - x}} + \frac{5 \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{1 - x}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{1 - x}} - \frac{\sqrt{x + 1}}{\sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07302, size = 59, normalized size = 1.23 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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