Integrand size = 17, antiderivative size = 48 \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=\frac {1}{2} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{3} (1-x)^{3/2} (1+x)^{3/2}+\frac {\arcsin (x)}{2} \]
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Time = 0.00 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=\frac {\arcsin (x)}{2}+\frac {1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac {1}{2} \sqrt {1-x} x \sqrt {x+1} \]
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=\frac {1}{6} \left (2+3 x-2 x^2\right ) \sqrt {1-x^2}+\arctan \left (\frac {\sqrt {1-x^2}}{1-x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(34)=68\).
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 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 | \(\frac {\left (2 x^{2}-3 x -2\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(77\) |
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Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=-\frac {1}{6} \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \sqrt {x + 1} \sqrt {-x + 1} - \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 6.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.48 \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=\begin {cases} - i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} + \frac {11 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {17 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} - \frac {11 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {17 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {\sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.58 \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04 \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=-\frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int (1-x)^{3/2} \sqrt {1+x} \, dx=\int {\left (1-x\right )}^{3/2}\,\sqrt {x+1} \,d x \]
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