Integrand size = 17, antiderivative size = 68 \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5 \arcsin (x)}{8} \]
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Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\frac {5 \arcsin (x)}{8}+\frac {1}{4} (x+1)^{3/2} (1-x)^{5/2}+\frac {5}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac {5}{8} x \sqrt {x+1} \sqrt {1-x} \]
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{4} \int (1-x)^{3/2} \sqrt {1+x} \, dx \\ & = \frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \sin ^{-1}(x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\frac {1}{24} \sqrt {1-x^2} \left (16+9 x-16 x^2+6 x^3\right )-\frac {5}{4} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {\left (6 x^{3}-16 x^{2}+9 x +16\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(82\) |
default | \(\frac {\left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}{4}+\frac {5 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{12}+\frac {5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(85\) |
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\frac {1}{24} \, {\left (6 \, x^{3} - 16 \, x^{2} + 9 \, x + 16\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 13.86 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.18 \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\begin {cases} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x - 1}} - \frac {23 i \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {x - 1}} + \frac {127 i \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {x - 1}} - \frac {133 i \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} + \frac {23 \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {1 - x}} - \frac {127 \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {1 - x}} + \frac {133 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} - \frac {5 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=-\frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {5}{8} \, \sqrt {-x^{2} + 1} x + \frac {5}{8} \, \arcsin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (48) = 96\).
Time = 0.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.49 \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {5}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int (1-x)^{5/2} \sqrt {1+x} \, dx=\int {\left (1-x\right )}^{5/2}\,\sqrt {x+1} \,d x \]
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