Integrand size = 17, antiderivative size = 82 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{35 \sqrt {1-x} \sqrt {1+x}} \]
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Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 39} \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {8 x}{35 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{35 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{35 (1-x)^{5/2} \sqrt {x+1}}+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}} \]
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Rule 39
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{7} \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {12}{35} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8}{35} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{35 \sqrt {1-x} \sqrt {1+x}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {13-4 x-20 x^2+24 x^3-8 x^4}{35 (1-x)^{7/2} \sqrt {1+x}} \]
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Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {8 x^{4}-24 x^{3}+20 x^{2}+4 x -13}{35 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}\) | \(35\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{4}-24 x^{3}+20 x^{2}+4 x -13\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(61\) |
default | \(\frac {1}{7 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}+\frac {4}{35 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {4}{35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {8}{35 \sqrt {1-x}\, \sqrt {1+x}}-\frac {8 \sqrt {1-x}}{35 \sqrt {1+x}}\) | \(72\) |
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {13 \, x^{5} - 39 \, x^{4} + 26 \, x^{3} + 26 \, x^{2} - {\left (8 \, x^{4} - 24 \, x^{3} + 20 \, x^{2} + 4 \, x - 13\right )} \sqrt {x + 1} \sqrt {-x + 1} - 39 \, x + 13}{35 \, {\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} - 3 \, x + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 38.90 (sec) , antiderivative size = 425, normalized size of antiderivative = 5.18 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (58) = 116\).
Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {8 \, x}{35 \, \sqrt {-x^{2} + 1}} - \frac {1}{7 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]
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Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{32 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{32 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left ({\left (93 \, x - 523\right )} {\left (x + 1\right )} + 1400\right )} {\left (x + 1\right )} - 1120\right )} \sqrt {x + 1} \sqrt {-x + 1}}{560 \, {\left (x - 1\right )}^{4}} \]
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Time = 0.39 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=-\frac {4\,x\,\sqrt {1-x}-13\,\sqrt {1-x}+20\,x^2\,\sqrt {1-x}-24\,x^3\,\sqrt {1-x}+8\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \]
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