Integrand size = 17, antiderivative size = 61 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=\frac {(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{63 (1-x)^{7/2}}+\frac {2 (1+x)^{5/2}}{315 (1-x)^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=\frac {2 (x+1)^{5/2}}{315 (1-x)^{5/2}}+\frac {2 (x+1)^{5/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{5/2}}{9 (1-x)^{9/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac {2}{9} \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{63 (1-x)^{7/2}}+\frac {2}{63} \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx \\ & = \frac {(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{63 (1-x)^{7/2}}+\frac {2 (1+x)^{5/2}}{315 (1-x)^{5/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=\frac {(1+x)^{5/2} \left (47-14 x+2 x^2\right )}{315 (1-x)^{9/2}} \]
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Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {5}{2}} \left (2 x^{2}-14 x +47\right )}{315 \left (1-x \right )^{\frac {9}{2}}}\) | \(25\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{5}-8 x^{4}+11 x^{3}+101 x^{2}+127 x +47\right )}{315 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(66\) |
default | \(\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (1-x \right )^{\frac {9}{2}}}-\frac {2 \sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}+\frac {\sqrt {1+x}}{63 \left (1-x \right )^{\frac {7}{2}}}+\frac {\sqrt {1+x}}{105 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{315 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{315 \sqrt {1-x}}\) | \(86\) |
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Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.41 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=\frac {47 \, x^{5} - 235 \, x^{4} + 470 \, x^{3} - 470 \, x^{2} - {\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} + 80 \, x + 47\right )} \sqrt {x + 1} \sqrt {-x + 1} + 235 \, x - 47}{315 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \]
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Result contains complex when optimal does not.
Time = 67.40 (sec) , antiderivative size = 675, normalized size of antiderivative = 11.07 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=\begin {cases} - \frac {2 i \left (x + 1\right )^{\frac {11}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} + \frac {22 i \left (x + 1\right )^{\frac {9}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} - \frac {99 i \left (x + 1\right )^{\frac {7}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} + \frac {126 i \left (x + 1\right )^{\frac {5}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {2 \left (x + 1\right )^{\frac {11}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} - \frac {22 \left (x + 1\right )^{\frac {9}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} + \frac {99 \left (x + 1\right )^{\frac {7}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} - \frac {126 \left (x + 1\right )^{\frac {5}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (43) = 86\).
Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.82 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{105 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x - 1\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=-\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 8\right )} + 63\right )} {\left (x + 1\right )}^{\frac {5}{2}} \sqrt {-x + 1}}{315 \, {\left (x - 1\right )}^{5}} \]
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Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.31 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {16\,x\,\sqrt {x+1}}{63}+\frac {47\,\sqrt {x+1}}{315}+\frac {x^2\,\sqrt {x+1}}{15}-\frac {2\,x^3\,\sqrt {x+1}}{63}+\frac {2\,x^4\,\sqrt {x+1}}{315}\right )}{x^5-5\,x^4+10\,x^3-10\,x^2+5\,x-1} \]
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