Integrand size = 17, antiderivative size = 109 \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9 \arcsin (x)}{16} \]
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Time = 0.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\frac {9 \arcsin (x)}{16}+\frac {1}{7} (x+1)^{5/2} (1-x)^{9/2}+\frac {3}{14} (x+1)^{5/2} (1-x)^{7/2}+\frac {3}{10} (x+1)^{5/2} (1-x)^{5/2}+\frac {3}{8} x (x+1)^{3/2} (1-x)^{3/2}+\frac {9}{16} x \sqrt {x+1} \sqrt {1-x} \]
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{7} \int (1-x)^{7/2} (1+x)^{3/2} \, dx \\ & = \frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {3}{2} \int (1-x)^{5/2} (1+x)^{3/2} \, dx \\ & = \frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {3}{2} \int (1-x)^{3/2} (1+x)^{3/2} \, dx \\ & = \frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \sin ^{-1}(x) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\frac {1}{560} \sqrt {1-x^2} \left (368+245 x-656 x^2+350 x^3+208 x^4-280 x^5+80 x^6\right )-\frac {9}{8} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\left (80 x^{6}-280 x^{5}+208 x^{4}+350 x^{3}-656 x^{2}+245 x +368\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{560 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {9 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(97\) |
default | \(\frac {\left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {5}{2}}}{7}+\frac {3 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {5}{2}}}{14}+\frac {3 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {5}{2}}}{10}+\frac {3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {5}{2}}}{8}+\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{8}-\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{16}-\frac {9 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {9 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(127\) |
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Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.61 \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\frac {1}{560} \, {\left (80 \, x^{6} - 280 \, x^{5} + 208 \, x^{4} + 350 \, x^{3} - 656 \, x^{2} + 245 \, x + 368\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {9}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Timed out. \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\frac {1}{7} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x^{2} - \frac {1}{2} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {23}{35} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {3}{8} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {9}{16} \, \sqrt {-x^{2} + 1} x + \frac {9}{16} \, \arcsin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (77) = 154\).
Time = 0.36 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.17 \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\frac {1}{1680} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{120} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{6} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {9}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int (1-x)^{9/2} (1+x)^{3/2} \, dx=\int {\left (1-x\right )}^{9/2}\,{\left (x+1\right )}^{3/2} \,d x \]
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