Optimal. Leaf size=70 \[ \frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1}+\frac {5}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {38, 41, 216} \[ \frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1}+\frac {5}{16} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 216
Rubi steps
\begin {align*} \int (1-x)^{5/2} (1+x)^{5/2} \, dx &=\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*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.49 \[ \frac {1}{48} \left (x \sqrt {1-x^2} \left (8 x^4-26 x^2+33\right )+15 \sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 51, normalized size = 0.73 \[ \frac {1}{48} \, {\left (8 \, x^{5} - 26 \, x^{3} + 33 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.31, size = 185, normalized size = 2.64 \[ \frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 113, normalized size = 1.61 \[ \frac {5 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{16 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {7}{2}}}{6}+\frac {\left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {7}{2}}}{6}+\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {7}{2}}}{8}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{24}-\frac {5 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{48}-\frac {5 \sqrt {-x +1}\, \sqrt {x +1}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.10, size = 41, normalized size = 0.59 \[ \frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {5}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {5}{16} \, \sqrt {-x^{2} + 1} x + \frac {5}{16} \, \arcsin \relax (x) \]
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 \[ \int {\left (1-x\right )}^{5/2}\,{\left (x+1\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 25.76, size = 286, normalized size = 4.09 \[ \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {i \left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {x - 1}} - \frac {7 i \left (x + 1\right )^{\frac {11}{2}}}{6 \sqrt {x - 1}} + \frac {67 i \left (x + 1\right )^{\frac {9}{2}}}{24 \sqrt {x - 1}} - \frac {55 i \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {\left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {1 - x}} + \frac {7 \left (x + 1\right )^{\frac {11}{2}}}{6 \sqrt {1 - x}} - \frac {67 \left (x + 1\right )^{\frac {9}{2}}}{24 \sqrt {1 - x}} + \frac {55 \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {5 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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