Integrand size = 17, antiderivative size = 70 \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5 \arcsin (x)}{16} \]
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Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {38, 41, 222} \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\frac {5 \arcsin (x)}{16}+\frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1} \]
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Rule 38
Rule 41
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{6} \int (1-x)^{3/2} (1+x)^{3/2} \, dx \\ & = \frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \sin ^{-1}(x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.73 \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\frac {1}{48} x \sqrt {1-x^2} \left (33-26 x^2+8 x^4\right )-\frac {5}{8} \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.14
method | result | size |
risch | \(-\frac {x \left (8 x^{4}-26 x^{2}+33\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{48 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(80\) |
default | \(\frac {\left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{6}+\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{6}+\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{8}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{24}-\frac {5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{48}-\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(113\) |
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.73 \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\frac {1}{48} \, {\left (8 \, x^{5} - 26 \, x^{3} + 33 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Timed out. \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {5}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {5}{16} \, \sqrt {-x^{2} + 1} x + \frac {5}{16} \, \arcsin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (50) = 100\).
Time = 0.35 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.64 \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int (1-x)^{5/2} (1+x)^{5/2} \, dx=\int {\left (1-x\right )}^{5/2}\,{\left (x+1\right )}^{5/2} \,d x \]
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