Integrand size = 17, antiderivative size = 41 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=\frac {(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{7/2}}{63 (1-x)^{7/2}} \]
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Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=\frac {(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac {1}{9} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{7/2}}{63 (1-x)^{7/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=\frac {(8-x) (1+x)^{7/2}}{63 (1-x)^{9/2}} \]
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Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (x -8\right ) \left (1+x \right )^{\frac {7}{2}}}{63 \left (1-x \right )^{\frac {9}{2}}}\) | \(18\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{5}-4 x^{4}-26 x^{3}-44 x^{2}-31 x -8\right )}{63 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(64\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{2 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{6 \left (1-x \right )^{\frac {9}{2}}}+\frac {5 \sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{126 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{42 \left (1-x \right )^{\frac {5}{2}}}-\frac {\sqrt {1+x}}{63 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{63 \sqrt {1-x}}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.02 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=\frac {8 \, x^{5} - 40 \, x^{4} + 80 \, x^{3} - 80 \, x^{2} + {\left (x^{4} - 5 \, x^{3} - 21 \, x^{2} - 23 \, x - 8\right )} \sqrt {x + 1} \sqrt {-x + 1} + 40 \, x - 8}{63 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \]
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Result contains complex when optimal does not.
Time = 71.90 (sec) , antiderivative size = 280, normalized size of antiderivative = 6.83 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=\begin {cases} \frac {i \left (x + 1\right )^{\frac {9}{2}}}{63 \sqrt {x - 1} \left (x + 1\right )^{4} - 504 \sqrt {x - 1} \left (x + 1\right )^{3} + 1512 \sqrt {x - 1} \left (x + 1\right )^{2} - 2016 \sqrt {x - 1} \left (x + 1\right ) + 1008 \sqrt {x - 1}} - \frac {9 i \left (x + 1\right )^{\frac {7}{2}}}{63 \sqrt {x - 1} \left (x + 1\right )^{4} - 504 \sqrt {x - 1} \left (x + 1\right )^{3} + 1512 \sqrt {x - 1} \left (x + 1\right )^{2} - 2016 \sqrt {x - 1} \left (x + 1\right ) + 1008 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {9}{2}}}{63 \sqrt {1 - x} \left (x + 1\right )^{4} - 504 \sqrt {1 - x} \left (x + 1\right )^{3} + 1512 \sqrt {1 - x} \left (x + 1\right )^{2} - 2016 \sqrt {1 - x} \left (x + 1\right ) + 1008 \sqrt {1 - x}} + \frac {9 \left (x + 1\right )^{\frac {7}{2}}}{63 \sqrt {1 - x} \left (x + 1\right )^{4} - 504 \sqrt {1 - x} \left (x + 1\right )^{3} + 1512 \sqrt {1 - x} \left (x + 1\right )^{2} - 2016 \sqrt {1 - x} \left (x + 1\right ) + 1008 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (29) = 58\).
Time = 0.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 5.32 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{2 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{126 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{42 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x - 1\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=\frac {{\left (x + 1\right )}^{\frac {7}{2}} {\left (x - 8\right )} \sqrt {-x + 1}}{63 \, {\left (x - 1\right )}^{5}} \]
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Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.95 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {23\,x\,\sqrt {x+1}}{63}+\frac {8\,\sqrt {x+1}}{63}+\frac {x^2\,\sqrt {x+1}}{3}+\frac {5\,x^3\,\sqrt {x+1}}{63}-\frac {x^4\,\sqrt {x+1}}{63}\right )}{x^5-5\,x^4+10\,x^3-10\,x^2+5\,x-1} \]
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