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 {1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac {1}{2} \sqrt {1-x} x \sqrt {x+1}+\frac {1}{2} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps
\begin {align*} \int (1-x)^{3/2} \sqrt {1+x} \, 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 = 0.06, size = 52, normalized size = 1.08 \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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 5.09, size = 129, normalized size = 2.69 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (\sqrt {1+x}-\frac {17 \left (1+x\right )^{\frac {3}{2}}}{6}-\text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}-\frac {\left (1+x\right )^{\frac {7}{2}}}{3}+\frac {11 \left (1+x\right )^{\frac {5}{2}}}{6}\right )}{\sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {11 \left (1+x\right )^{\frac {5}{2}}}{6 \sqrt {1-x}}-\frac {\sqrt {1+x}}{\sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {7}{2}}}{3 \sqrt {1-x}}+\frac {17 \left (1+x\right )^{\frac {3}{2}}}{6 \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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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 = 0.16, size = 71, normalized size = 1.48
method | result | size |
default | \(\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{3}+\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(71\) |
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 = 0.35, 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 = 0.30, 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 = 3.40, size = 167, normalized size = 3.48 \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 {11 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {17 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 {11 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {17 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (34) = 68\).
time = 0.01, size = 189, normalized size = 3.94 \begin {gather*} -2 \left (2 \left (\left (\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}-\frac {7}{24}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {3}{8}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )+4 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\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 {\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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