Optimal. Leaf size=47 \[ -\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \sin ^{-1}(x) \]
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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 = {52, 41, 222}
\begin {gather*} \frac {3 \text {ArcSin}(x)}{2}-\frac {1}{2} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{2} \sqrt {1-x} \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 222
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\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} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {1-x} \left (4+5 x+x^2\right )}{2 \sqrt {1+x}}-3 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 57, normalized size = 1.21
method | result | size |
default | \(-\frac {\left (1+x \right )^{\frac {3}{2}} \sqrt {1-x}}{2}-\frac {3 \sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {3 \arcsin \left (x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(57\) |
risch | \(\frac {\left (4+x \right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {3 \arcsin \left (x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, 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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 40, normalized size = 0.85 \begin {gather*} -\frac {1}{2} \, {\left (x + 4\right )} \sqrt {x + 1} \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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Sympy [C] Result contains complex when optimal does not.
time = 2.25, size = 134, normalized size = 2.85 \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 {i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\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 {\left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {3 \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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Giac [A]
time = 0.68, size = 31, normalized size = 0.66 \begin {gather*} -\frac {1}{2} \, {\left (x + 4\right )} \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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