Optimal. Leaf size=69 \[ -\frac {1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1}+\frac {3}{8} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ -\frac {1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1}+\frac {3}{8} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 49
Rule 216
Rubi steps
\begin {align*} \int (1-x)^{3/2} (1+x)^{5/2} \, dx &=-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.80 \[ \frac {1}{40} \left (\sqrt {1-x^2} \left (-8 x^4-10 x^3+16 x^2+25 x-8\right )-30 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 57, normalized size = 0.83 \[ -\frac {1}{40} \, {\left (8 \, x^{4} + 10 \, x^{3} - 16 \, x^{2} - 25 \, x + 8\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.18, size = 114, normalized size = 1.65 \[ -\frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {3}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 99, normalized size = 1.43 \[ \frac {3 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{8 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {7}{2}}}{5}+\frac {3 \sqrt {-x +1}\, \left (x +1\right )^{\frac {7}{2}}}{20}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{20}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{8}-\frac {3 \sqrt {-x +1}\, \sqrt {x +1}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 40, normalized size = 0.58 \[ -\frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {-x^{2} + 1} x + \frac {3}{8} \, \arcsin \relax (x) \]
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 \[ \int {\left (1-x\right )}^{3/2}\,{\left (x+1\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 16.53, size = 246, normalized size = 3.57 \[ \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {x - 1}} + \frac {19 i \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {x - 1}} - \frac {23 i \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {1 - x}} - \frac {19 \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {1 - x}} + \frac {23 \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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