Optimal. Leaf size=48 \[ \frac {1}{2} \sqrt {1-x} x \sqrt {1+x}-\frac {1}{3} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 = {51, 38, 41, 222}
\begin {gather*} \frac {\text {ArcSin}(x)}{2}-\frac {1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac {1}{2} \sqrt {1-x} x \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 51
Rule 222
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*}
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Mathematica [A]
time = 0.07, size = 54, normalized size = 1.12 \begin {gather*} \frac {\sqrt {1-x} \left (-2+x+5 x^2+2 x^3\right )}{6 \sqrt {1+x}}-\tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs.
\(2(34)=68\).
time = 0.14, size = 71, normalized size = 1.48
method | result | size |
default | \(\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{3}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{6}-\frac {\sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(71\) |
risch | \(-\frac {\left (2 x^{2}+3 x -2\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 28, normalized size = 0.58 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 47, normalized size = 0.98 \begin {gather*} \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 {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 = 4.55, size = 163, normalized size = 3.40 \begin {gather*} \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}\: \left |{x + 1}\right | > 2 \\\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 66, normalized size = 1.38 \begin {gather*} \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} + \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 \sqrt {1-x}\,{\left (x+1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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