Integrand size = 17, antiderivative size = 61 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}} \]
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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} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {2 (x+1)^{7/2}}{693 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2}{11} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2}{99} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {(1+x)^{7/2} \left (79-18 x+2 x^2\right )}{693 (1-x)^{11/2}} \]
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Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {7}{2}} \left (2 x^{2}-18 x +79\right )}{693 \left (1-x \right )^{\frac {11}{2}}}\) | \(25\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{6}-10 x^{5}+19 x^{4}+216 x^{3}+404 x^{2}+298 x +79\right )}{693 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(71\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{3 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{12 \left (1-x \right )^{\frac {11}{2}}}+\frac {5 \sqrt {1+x}}{22 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{396 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{231 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{693 \sqrt {1-x}}\) | \(114\) |
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (43) = 86\).
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.64 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {79 \, x^{6} - 474 \, x^{5} + 1185 \, x^{4} - 1580 \, x^{3} + 1185 \, x^{2} + {\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} + 185 \, x^{2} + 219 \, x + 79\right )} \sqrt {x + 1} \sqrt {-x + 1} - 474 \, x + 79}{693 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \]
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Result contains complex when optimal does not.
Time = 178.85 (sec) , antiderivative size = 784, normalized size of antiderivative = 12.85 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\begin {cases} \frac {2 i \left (x + 1\right )^{\frac {13}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} - \frac {26 i \left (x + 1\right )^{\frac {11}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} + \frac {143 i \left (x + 1\right )^{\frac {9}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} - \frac {198 i \left (x + 1\right )^{\frac {7}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {2 \left (x + 1\right )^{\frac {13}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} + \frac {26 \left (x + 1\right )^{\frac {11}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} - \frac {143 \left (x + 1\right )^{\frac {9}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} + \frac {198 \left (x + 1\right )^{\frac {7}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (43) = 86\).
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.41 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{3 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{12 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{22 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{396 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{231 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x - 1\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 10\right )} + 99\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{693 \, {\left (x - 1\right )}^{6}} \]
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Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {73\,x\,\sqrt {x+1}}{231}+\frac {79\,\sqrt {x+1}}{693}+\frac {185\,x^2\,\sqrt {x+1}}{693}+\frac {31\,x^3\,\sqrt {x+1}}{693}-\frac {4\,x^4\,\sqrt {x+1}}{231}+\frac {2\,x^5\,\sqrt {x+1}}{693}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \]
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