Optimal. Leaf size=89 \[ \frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac {7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac {7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac {7}{16} x \sqrt {x+1} \sqrt {1-x}+\frac {7}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac {7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac {7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac {7}{16} x \sqrt {x+1} \sqrt {1-x}+\frac {7}{16} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 49
Rule 216
Rubi steps
\begin {align*} \int (1-x)^{7/2} (1+x)^{3/2} \, dx &=\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*}
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Mathematica [A] time = 0.06, size = 61, normalized size = 0.69 \[ \frac {1}{240} \sqrt {1-x^2} \left (-40 x^5+96 x^4+10 x^3-192 x^2+135 x+96\right )-\frac {7}{8} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 62, normalized size = 0.70 \[ -\frac {1}{240} \, {\left (40 \, x^{5} - 96 \, x^{4} - 10 \, x^{3} + 192 \, x^{2} - 135 \, x - 96\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.33, size = 185, normalized size = 2.08 \[ -\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 113, normalized size = 1.27 \[ \frac {7 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{16 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {7}{2}} \left (x +1\right )^{\frac {5}{2}}}{6}+\frac {7 \left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {5}{2}}}{30}+\frac {7 \left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {5}{2}}}{24}+\frac {7 \sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{24}-\frac {7 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{48}-\frac {7 \sqrt {-x +1}\, \sqrt {x +1}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 52, normalized size = 0.58 \[ -\frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {2}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {7}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {7}{16} \, \sqrt {-x^{2} + 1} x + \frac {7}{16} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (1-x\right )}^{7/2}\,{\left (x+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 32.99, size = 289, normalized size = 3.25 \[ \begin {cases} - \frac {7 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {i \left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {x - 1}} + \frac {47 i \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {x - 1}} - \frac {683 i \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {x - 1}} + \frac {1151 i \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {x - 1}} - \frac {1543 i \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {x - 1}} - \frac {7 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {7 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {7 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {\left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {1 - x}} - \frac {47 \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {1 - x}} + \frac {683 \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {1 - x}} - \frac {1151 \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {1 - x}} + \frac {1543 \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {1 - x}} + \frac {7 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {7 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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