Integrand size = 17, antiderivative size = 102 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{63 \sqrt {1-x} \sqrt {1+x}} \]
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Time = 0.01 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 39} \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {8 x}{63 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{63 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{63 (1-x)^{5/2} \sqrt {x+1}}+\frac {5}{63 (1-x)^{7/2} \sqrt {x+1}}+\frac {1}{9 (1-x)^{9/2} \sqrt {x+1}} \]
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Rule 39
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{9} \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {20}{63} \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{21} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8}{63} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{63 \sqrt {1-x} \sqrt {1+x}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {20-17 x-16 x^2+44 x^3-32 x^4+8 x^5}{63 (1-x)^{9/2} \sqrt {1+x}} \]
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Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.39
method | result | size |
gosper | \(\frac {8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20}{63 \sqrt {1+x}\, \left (1-x \right )^{\frac {9}{2}}}\) | \(40\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20\right )}{63 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(66\) |
default | \(\frac {1}{9 \left (1-x \right )^{\frac {9}{2}} \sqrt {1+x}}+\frac {5}{63 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}+\frac {4}{63 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {4}{63 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {8}{63 \sqrt {1-x}\, \sqrt {1+x}}-\frac {8 \sqrt {1-x}}{63 \sqrt {1+x}}\) | \(86\) |
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Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {20 \, x^{6} - 80 \, x^{5} + 100 \, x^{4} - 100 \, x^{2} - {\left (8 \, x^{5} - 32 \, x^{4} + 44 \, x^{3} - 16 \, x^{2} - 17 \, x + 20\right )} \sqrt {x + 1} \sqrt {-x + 1} + 80 \, x - 20}{63 \, {\left (x^{6} - 4 \, x^{5} + 5 \, x^{4} - 5 \, x^{2} + 4 \, x - 1\right )}} \]
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Result contains complex when optimal does not.
Time = 113.36 (sec) , antiderivative size = 593, normalized size of antiderivative = 5.81 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {72 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {252 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {315 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {63 \sqrt {-1 + \frac {2}{x + 1}}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{5}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {72 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {252 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {315 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {63 i \sqrt {1 - \frac {2}{x + 1}}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (72) = 144\).
Time = 0.22 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {8 \, x}{63 \, \sqrt {-x^{2} + 1}} + \frac {1}{9 \, {\left (\sqrt {-x^{2} + 1} x^{4} - 4 \, \sqrt {-x^{2} + 1} x^{3} + 6 \, \sqrt {-x^{2} + 1} x^{2} - 4 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {5}{63 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{63 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{63 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{64 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{64 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left ({\left ({\left (193 \, x - 1481\right )} {\left (x + 1\right )} + 5544\right )} {\left (x + 1\right )} - 8400\right )} {\left (x + 1\right )} + 5040\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2016 \, {\left (x - 1\right )}^{5}} \]
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Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx=\frac {17\,x\,\sqrt {1-x}-20\,\sqrt {1-x}+16\,x^2\,\sqrt {1-x}-44\,x^3\,\sqrt {1-x}+32\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{63\,{\left (x-1\right )}^5\,\sqrt {x+1}} \]
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