Integrand size = 17, antiderivative size = 67 \[ \int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx=-\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5 \arcsin (x)}{2} \]
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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} \[ \int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx=\frac {5 \arcsin (x)}{2}-\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {5}{6} \sqrt {1-x} (x+1)^{3/2}-\frac {5}{2} \sqrt {1-x} \sqrt {x+1} \]
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Rule 41
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{3} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx \\ & = -\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx \\ & = -\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\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} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \sin ^{-1}(x) \\ \end{align*}
Time = 0.07 (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 = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{3}-\frac {5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{6}-\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(71\) |
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 = 11.19 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.54 \[ \int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx=\begin {cases} - 5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {5 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (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 = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58 \[ \int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx=-\frac {1}{6} \, {\left ({\left (2 \, x + 7\right )} {\left (x + 1\right )} + 15\right )} \sqrt {x + 1} \sqrt {-x + 1} + 5 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx=\int \frac {{\left (x+1\right )}^{5/2}}{\sqrt {1-x}} \,d x \]
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