Integrand size = 17, antiderivative size = 89 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\frac {7}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7 \arcsin (x)}{16} \]
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Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\frac {7 \arcsin (x)}{16}+\frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac {7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac {7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac {7}{16} x \sqrt {x+1} \sqrt {1-x} \]
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7}{6} \int (1-x)^{5/2} (1+x)^{3/2} \, dx \\ & = \frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7}{6} \int (1-x)^{3/2} (1+x)^{3/2} \, dx \\ & = \frac {7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {7}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {7}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {7}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7}{16} \sin ^{-1}(x) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\frac {1}{240} \sqrt {1-x^2} \left (96+135 x-192 x^2+10 x^3+96 x^4-40 x^5\right )-\frac {7}{8} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {\left (40 x^{5}-96 x^{4}-10 x^{3}+192 x^{2}-135 x -96\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{240 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {7 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(92\) |
default | \(\frac {\left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {5}{2}}}{6}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {5}{2}}}{30}+\frac {7 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {5}{2}}}{24}+\frac {7 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{24}-\frac {7 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{48}-\frac {7 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {7 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(113\) |
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Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=-\frac {1}{240} \, {\left (40 \, x^{5} - 96 \, x^{4} - 10 \, x^{3} + 192 \, x^{2} - 135 \, x - 96\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 164.06 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.22 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\begin {cases} - \frac {7 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {i \left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {x - 1}} + \frac {47 i \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {x - 1}} - \frac {683 i \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {x - 1}} + \frac {1151 i \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {x - 1}} - \frac {1543 i \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {x - 1}} - \frac {7 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {7 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {7 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {\left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {1 - x}} - \frac {47 \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {1 - x}} + \frac {683 \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {1 - x}} - \frac {1151 \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {1 - x}} + \frac {1543 \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {1 - x}} + \frac {7 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {7 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=-\frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {2}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {7}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {7}{16} \, \sqrt {-x^{2} + 1} x + \frac {7}{16} \, \arcsin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (63) = 126\).
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.08 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=-\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\int {\left (1-x\right )}^{7/2}\,{\left (x+1\right )}^{3/2} \,d x \]
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