Optimal. Leaf size=67 \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {5}{6} \sqrt {1-x} (x+1)^{3/2}-\frac {5}{2} \sqrt {1-x} \sqrt {x+1}+\frac {5}{2} \sin ^{-1}(x) \]
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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} \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {5}{6} \sqrt {1-x} (x+1)^{3/2}-\frac {5}{2} \sqrt {1-x} \sqrt {x+1}+\frac {5}{2} \sin ^{-1}(x) \]
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} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{3} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\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} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.66 \[ -\frac {1}{6} \sqrt {1-x^2} \left (2 x^2+9 x+22\right )-5 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, 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 = 1.05, size = 39, normalized size = 0.58 \[ -\frac {1}{6} \, {\left ({\left (2 \, x + 7\right )} {\left (x + 1\right )} + 15\right )} \sqrt {x + 1} \sqrt {-x + 1} + 5 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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 \[ \frac {5 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{3}-\frac {5 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{6}-\frac {5 \sqrt {-x +1}\, \sqrt {x +1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, 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 \[ \int \frac {{\left (x+1\right )}^{5/2}}{\sqrt {1-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.50, size = 172, normalized size = 2.57 \[ \begin {cases} - 5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {5 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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