Integrand size = 17, antiderivative size = 101 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac {8 (1+x)^{5/2}}{15015 (1-x)^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 101, 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, 37} \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {8 (x+1)^{5/2}}{15015 (1-x)^{5/2}}+\frac {8 (x+1)^{5/2}}{3003 (1-x)^{7/2}}+\frac {4 (x+1)^{5/2}}{429 (1-x)^{9/2}}+\frac {4 (x+1)^{5/2}}{143 (1-x)^{11/2}}+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4}{13} \int \frac {(1+x)^{3/2}}{(1-x)^{13/2}} \, dx \\ & = \frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {12}{143} \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx \\ & = \frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8}{429} \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac {8 \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx}{3003} \\ & = \frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac {8 (1+x)^{5/2}}{15015 (1-x)^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {(1+x)^{5/2} \left (1763-852 x+308 x^2-72 x^3+8 x^4\right )}{15015 (1-x)^{13/2}} \]
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Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {5}{2}} \left (8 x^{4}-72 x^{3}+308 x^{2}-852 x +1763\right )}{15015 \left (1-x \right )^{\frac {13}{2}}}\) | \(35\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{7}-48 x^{6}+116 x^{5}-136 x^{4}+59 x^{3}+3041 x^{2}+4437 x +1763\right )}{15015 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{6} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(76\) |
default | \(\frac {\left (1+x \right )^{\frac {3}{2}}}{5 \left (1-x \right )^{\frac {13}{2}}}-\frac {6 \sqrt {1+x}}{65 \left (1-x \right )^{\frac {13}{2}}}+\frac {3 \sqrt {1+x}}{715 \left (1-x \right )^{\frac {11}{2}}}+\frac {\sqrt {1+x}}{429 \left (1-x \right )^{\frac {9}{2}}}+\frac {4 \sqrt {1+x}}{3003 \left (1-x \right )^{\frac {7}{2}}}+\frac {4 \sqrt {1+x}}{5005 \left (1-x \right )^{\frac {5}{2}}}+\frac {8 \sqrt {1+x}}{15015 \left (1-x \right )^{\frac {3}{2}}}+\frac {8 \sqrt {1+x}}{15015 \sqrt {1-x}}\) | \(114\) |
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {1763 \, x^{7} - 12341 \, x^{6} + 37023 \, x^{5} - 61705 \, x^{4} + 61705 \, x^{3} - 37023 \, x^{2} - {\left (8 \, x^{6} - 56 \, x^{5} + 172 \, x^{4} - 308 \, x^{3} + 367 \, x^{2} + 2674 \, x + 1763\right )} \sqrt {x + 1} \sqrt {-x + 1} + 12341 \, x - 1763}{15015 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} \]
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Timed out. \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (71) = 142\).
Time = 0.22 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.66 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{5 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {6 \, \sqrt {-x^{2} + 1}}{65 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {3 \, \sqrt {-x^{2} + 1}}{715 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{429 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{5005 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x - 1\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=-\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 12\right )} + 143\right )} {\left (x + 1\right )} - 429\right )} {\left (x + 1\right )} + 3003\right )} {\left (x + 1\right )}^{\frac {5}{2}} \sqrt {-x + 1}}{15015 \, {\left (x - 1\right )}^{7}} \]
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Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {382\,x\,\sqrt {x+1}}{2145}+\frac {1763\,\sqrt {x+1}}{15015}+\frac {367\,x^2\,\sqrt {x+1}}{15015}-\frac {4\,x^3\,\sqrt {x+1}}{195}+\frac {172\,x^4\,\sqrt {x+1}}{15015}-\frac {8\,x^5\,\sqrt {x+1}}{2145}+\frac {8\,x^6\,\sqrt {x+1}}{15015}\right )}{x^7-7\,x^6+21\,x^5-35\,x^4+35\,x^3-21\,x^2+7\,x-1} \]
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