Optimal. Leaf size=61 \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0113053, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {80, 50, 41, 216} \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{x (1+x)^{3/2}}{\sqrt{1-x}} \, dx &=-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{2}{3} \int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0241145, size = 42, normalized size = 0.69 \[ -\frac{1}{3} \sqrt{1-x^2} \left (x^2+3 x+5\right )-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{3}\sqrt{1-x}\sqrt{1+x} \left ( -{x}^{2}\sqrt{-{x}^{2}+1}-3\,x\sqrt{-{x}^{2}+1}+3\,\arcsin \left ( x \right ) -5\,\sqrt{-{x}^{2}+1} \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 = 1.98909, size = 54, normalized size = 0.89 \begin{align*} -\frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \sqrt{-x^{2} + 1} x - \frac{5}{3} \, \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78847, size = 124, normalized size = 2.03 \begin{align*} -\frac{1}{3} \,{\left (x^{2} + 3 \, x + 5\right )} \sqrt{x + 1} \sqrt{-x + 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 [A] time = 59.4705, size = 129, normalized size = 2.11 \begin{align*} - 2 \left (\begin{cases} - \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} - \sqrt{1 - x} \sqrt{x + 1} + \frac{3 \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{3 x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{\left (1 - x\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} - 2 \sqrt{1 - x} \sqrt{x + 1} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29544, size = 50, normalized size = 0.82 \begin{align*} -\frac{1}{3} \,{\left ({\left (x + 2\right )}{\left (x + 1\right )} + 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2 \, \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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