Optimal. Leaf size=47 \[ \frac {1}{2} \sqrt {x+1} (1-x)^{3/2}+\frac {3}{2} \sqrt {x+1} \sqrt {1-x}+\frac {3}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {50, 41, 216} \begin {gather*} \frac {1}{2} \sqrt {x+1} (1-x)^{3/2}+\frac {3}{2} \sqrt {x+1} \sqrt {1-x}+\frac {3}{2} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx &=\frac {1}{2} (1-x)^{3/2} \sqrt {1+x}+\frac {3}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=\frac {3}{2} \sqrt {1-x} \sqrt {1+x}+\frac {1}{2} (1-x)^{3/2} \sqrt {1+x}+\frac {3}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {3}{2} \sqrt {1-x} \sqrt {1+x}+\frac {1}{2} (1-x)^{3/2} \sqrt {1+x}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{2} \sqrt {1-x} \sqrt {1+x}+\frac {1}{2} (1-x)^{3/2} \sqrt {1+x}+\frac {3}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x+1} \left (x^2-5 x+4\right )}{2 \sqrt {1-x}}-3 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 67, normalized size = 1.43 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {3 (x+1)}{1-x}+5\right )}{\sqrt {1-x} \left (\frac {x+1}{1-x}+1\right )^2}+3 \tan ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 40, normalized size = 0.85 \begin {gather*} -\frac {1}{2} \, \sqrt {x + 1} {\left (x - 4\right )} \sqrt {-x + 1} - 3 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 44, normalized size = 0.94 \begin {gather*} -\frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + 3 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 57, normalized size = 1.21 \begin {gather*} \frac {3 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {3}{2}} \sqrt {x +1}}{2}+\frac {3 \sqrt {-x +1}\, \sqrt {x +1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 28, normalized size = 0.60 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} x + 2 \, \sqrt {-x^{2} + 1} + \frac {3}{2} \, \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (1-x\right )}^{3/2}}{\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.59, size = 139, normalized size = 2.96 \begin {gather*} \begin {cases} - 3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} + \frac {7 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} - \frac {5 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {7 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} + \frac {5 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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