Integrand size = 17, antiderivative size = 63 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \arcsin (x) \]
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Time = 0.01 (sec) , antiderivative size = 63, 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 = {49, 52, 41, 222} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=5 \arcsin (x)+\frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {10 (x+1)^{3/2}}{3 \sqrt {1-x}}-5 \sqrt {1-x} \sqrt {x+1} \]
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Rule 41
Rule 49
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}-\frac {5}{3} \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx \\ & = -\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx \\ & = -5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \sin ^{-1}(x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {\sqrt {1-x^2} \left (23-34 x+3 x^2\right )}{3 (-1+x)^2}-10 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\frac {\left (3 x^{3}-31 x^{2}-11 x +23\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(84\) |
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Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {23 \, x^{2} + {\left (3 \, x^{2} - 34 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 46 \, x + 23}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 5.62 (sec) , antiderivative size = 575, normalized size of antiderivative = 9.13 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=\begin {cases} - \frac {30 i \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {15 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 i \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {30 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {3 i \left (x + 1\right )^{15}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {40 i \left (x + 1\right )^{14}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 i \left (x + 1\right )^{13}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {30 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {3 \left (x + 1\right )^{15}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {40 \left (x + 1\right )^{14}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 \left (x + 1\right )^{13}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (47) = 94\).
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.57 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {10 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {35 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + 5 \, \arcsin \left (x\right ) \]
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Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {{\left ({\left (3 \, x - 37\right )} {\left (x + 1\right )} + 60\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 10 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=\int \frac {{\left (x+1\right )}^{5/2}}{{\left (1-x\right )}^{5/2}} \,d x \]
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