Integrand size = 17, antiderivative size = 81 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=\frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{3003 (1-x)^{7/2}} \]
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Time = 0.01 (sec) , antiderivative size = 81, 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, 37} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=\frac {2 (x+1)^{7/2}}{3003 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{429 (1-x)^{9/2}}+\frac {3 (x+1)^{7/2}}{143 (1-x)^{11/2}}+\frac {(x+1)^{7/2}}{13 (1-x)^{13/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3}{13} \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {6}{143} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac {2}{429} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{3003 (1-x)^{7/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=\frac {(1+x)^{7/2} \left (310-97 x+20 x^2-2 x^3\right )}{3003 (1-x)^{13/2}} \]
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Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.37
method | result | size |
gosper | \(-\frac {\left (1+x \right )^{\frac {7}{2}} \left (2 x^{3}-20 x^{2}+97 x -310\right )}{3003 \left (1-x \right )^{\frac {13}{2}}}\) | \(30\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{7}-12 x^{6}+29 x^{5}-34 x^{4}-736 x^{3}-1492 x^{2}-1143 x -310\right )}{3003 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{6} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(76\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{4 \left (1-x \right )^{\frac {13}{2}}}-\frac {\left (1+x \right )^{\frac {3}{2}}}{4 \left (1-x \right )^{\frac {13}{2}}}+\frac {3 \sqrt {1+x}}{26 \left (1-x \right )^{\frac {13}{2}}}-\frac {3 \sqrt {1+x}}{572 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{1716 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{3003 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{1001 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{3003 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{3003 \sqrt {1-x}}\) | \(128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (57) = 114\).
Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.42 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=\frac {310 \, x^{7} - 2170 \, x^{6} + 6510 \, x^{5} - 10850 \, x^{4} + 10850 \, x^{3} - 6510 \, x^{2} + {\left (2 \, x^{6} - 14 \, x^{5} + 43 \, x^{4} - 77 \, x^{3} - 659 \, x^{2} - 833 \, x - 310\right )} \sqrt {x + 1} \sqrt {-x + 1} + 2170 \, x - 310}{3003 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} \]
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Timed out. \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (57) = 114\).
Time = 0.22 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.01 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{4 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} - \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{4 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{26 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{572 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{1716 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{1001 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x - 1\right )}} \]
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none
Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=\frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 12\right )} + 143\right )} {\left (x + 1\right )} - 429\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{3003 \, {\left (x - 1\right )}^{7}} \]
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Time = 0.40 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.36 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {119\,x\,\sqrt {x+1}}{429}+\frac {310\,\sqrt {x+1}}{3003}+\frac {659\,x^2\,\sqrt {x+1}}{3003}+\frac {x^3\,\sqrt {x+1}}{39}-\frac {43\,x^4\,\sqrt {x+1}}{3003}+\frac {2\,x^5\,\sqrt {x+1}}{429}-\frac {2\,x^6\,\sqrt {x+1}}{3003}\right )}{x^7-7\,x^6+21\,x^5-35\,x^4+35\,x^3-21\,x^2+7\,x-1} \]
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