Integrand size = 17, antiderivative size = 87 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35 \arcsin (x)}{8} \]
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Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 41, 222} \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\frac {35 \arcsin (x)}{8}+\frac {1}{4} \sqrt {x+1} (1-x)^{7/2}+\frac {7}{12} \sqrt {x+1} (1-x)^{5/2}+\frac {35}{24} \sqrt {x+1} (1-x)^{3/2}+\frac {35}{8} \sqrt {x+1} \sqrt {1-x} \]
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Rule 41
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {7}{4} \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx \\ & = \frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{12} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx \\ & = \frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx \\ & = \frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \sin ^{-1}(x) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\frac {1}{24} \sqrt {1-x^2} \left (160-81 x+32 x^2-6 x^3\right )-\frac {35}{4} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {\left (6 x^{3}-32 x^{2}+81 x -160\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(82\) |
default | \(\frac {\left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}{4}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{12}+\frac {35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{24}+\frac {35 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(85\) |
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Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {1}{24} \, {\left (6 \, x^{3} - 32 \, x^{2} + 81 \, x - 160\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {35}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 20.72 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.26 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\begin {cases} - \frac {i \sqrt {x - 1} \left (x + 1\right )^{\frac {7}{2}}}{4} + \frac {25 i \sqrt {x - 1} \left (x + 1\right )^{\frac {5}{2}}}{12} - \frac {163 i \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}}{24} + \frac {93 i \sqrt {x - 1} \sqrt {x + 1}}{8} - \frac {35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} - \frac {31 \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {1 - x}} + \frac {263 \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {1 - x}} - \frac {605 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} + \frac {93 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} + \frac {4}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {27}{8} \, \sqrt {-x^{2} + 1} x + \frac {20}{3} \, \sqrt {-x^{2} + 1} + \frac {35}{8} \, \arcsin \left (x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {35}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\int \frac {{\left (1-x\right )}^{7/2}}{\sqrt {x+1}} \,d x \]
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