Optimal. Leaf size=67 \[ \frac {1}{3} \sqrt {x+1} (1-x)^{5/2}+\frac {5}{6} \sqrt {x+1} (1-x)^{3/2}+\frac {5}{2} \sqrt {x+1} \sqrt {1-x}+\frac {5}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 67, 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 = {50, 41, 216} \[ \frac {1}{3} \sqrt {x+1} (1-x)^{5/2}+\frac {5}{6} \sqrt {x+1} (1-x)^{3/2}+\frac {5}{2} \sqrt {x+1} \sqrt {1-x}+\frac {5}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx &=\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{3} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx\\ &=\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=\frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.81 \[ \frac {\sqrt {x+1} \left (-2 x^3+11 x^2-31 x+22\right )}{6 \sqrt {1-x}}-5 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 47, normalized size = 0.70 \[ \frac {1}{6} \, {\left (2 \, x^{2} - 9 \, x + 22\right )} \sqrt {x + 1} \sqrt {-x + 1} - 5 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 69, normalized size = 1.03 \[ \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + 5 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 71, normalized size = 1.06 \[ \frac {5 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {5}{2}} \sqrt {x +1}}{3}+\frac {5 \left (-x +1\right )^{\frac {3}{2}} \sqrt {x +1}}{6}+\frac {5 \sqrt {-x +1}\, \sqrt {x +1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 42, normalized size = 0.63 \[ \frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {3}{2} \, \sqrt {-x^{2} + 1} x + \frac {11}{3} \, \sqrt {-x^{2} + 1} + \frac {5}{2} \, \arcsin \relax (x) \]
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 \[ \int \frac {{\left (1-x\right )}^{5/2}}{\sqrt {x+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.64, size = 175, normalized size = 2.61 \[ \begin {cases} - 5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {17 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} + \frac {59 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} - \frac {11 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {17 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} - \frac {59 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} + \frac {11 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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