Optimal. Leaf size=67 \[ \frac{1}{3} \sqrt{x+1} (1-x)^{5/2}+\frac{5}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{5}{2} \sqrt{x+1} \sqrt{1-x}+\frac{5}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0106446, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 41, 216} \[ \frac{1}{3} \sqrt{x+1} (1-x)^{5/2}+\frac{5}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{5}{2} \sqrt{x+1} \sqrt{1-x}+\frac{5}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-x)^{5/2}}{\sqrt{1+x}} \, dx &=\frac{1}{3} (1-x)^{5/2} \sqrt{1+x}+\frac{5}{3} \int \frac{(1-x)^{3/2}}{\sqrt{1+x}} \, dx\\ &=\frac{5}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{1}{3} (1-x)^{5/2} \sqrt{1+x}+\frac{5}{2} \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx\\ &=\frac{5}{2} \sqrt{1-x} \sqrt{1+x}+\frac{5}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{1}{3} (1-x)^{5/2} \sqrt{1+x}+\frac{5}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{5}{2} \sqrt{1-x} \sqrt{1+x}+\frac{5}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{1}{3} (1-x)^{5/2} \sqrt{1+x}+\frac{5}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{5}{2} \sqrt{1-x} \sqrt{1+x}+\frac{5}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{1}{3} (1-x)^{5/2} \sqrt{1+x}+\frac{5}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0256524, size = 54, normalized size = 0.81 \[ \frac{\sqrt{x+1} \left (-2 x^3+11 x^2-31 x+22\right )}{6 \sqrt{1-x}}-5 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 71, normalized size = 1.1 \begin{align*}{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{5}{2}}}\sqrt{1+x}}+{\frac{5}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{5}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\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 = 1.50438, size = 57, normalized size = 0.85 \begin{align*} \frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{3}{2} \, \sqrt{-x^{2} + 1} x + \frac{11}{3} \, \sqrt{-x^{2} + 1} + \frac{5}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54617, size = 127, normalized size = 1.9 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} - 9 \, x + 22\right )} \sqrt{x + 1} \sqrt{-x + 1} - 5 \, \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 = 10.0033, size = 175, normalized size = 2.61 \begin{align*} \begin{cases} - 5 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{17 i \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{x - 1}} + \frac{59 i \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{x - 1}} - \frac{11 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{1 - x}} + \frac{17 \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{1 - x}} - \frac{59 \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{1 - x}} + \frac{11 \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 = 1.07833, size = 93, normalized size = 1.39 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x - 5\right )}{\left (x + 1\right )} + 9\right )} \sqrt{x + 1} \sqrt{-x + 1} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + 5 \, \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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