Integrand size = 17, antiderivative size = 67 \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5 \arcsin (x)}{2} \]
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Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 41, 222} \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\frac {5 \arcsin (x)}{2}+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}+\frac {5}{6} \sqrt {x+1} (1-x)^{3/2}+\frac {5}{2} \sqrt {x+1} \sqrt {1-x} \]
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Rule 41
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{3} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx \\ & = \frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx \\ & = \frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \sin ^{-1}(x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69 \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\frac {1}{6} \sqrt {1-x^2} \left (22-9 x+2 x^2\right )-5 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {\left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{3}+\frac {5 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{6}+\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(71\) |
risch | \(-\frac {\left (2 x^{2}-9 x +22\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(77\) |
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Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\frac {1}{6} \, {\left (2 \, x^{2} - 9 \, x + 22\right )} \sqrt {x + 1} \sqrt {-x + 1} - 5 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 5.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.58 \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\begin {cases} - 5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {17 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} + \frac {59 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} - \frac {11 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {17 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} - \frac {59 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} + \frac {11 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63 \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {3}{2} \, \sqrt {-x^{2} + 1} x + \frac {11}{3} \, \sqrt {-x^{2} + 1} + \frac {5}{2} \, \arcsin \left (x\right ) \]
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Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.03 \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + 5 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx=\int \frac {{\left (1-x\right )}^{5/2}}{\sqrt {x+1}} \,d x \]
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