Optimal. Leaf size=67 \[ \frac {5}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {1}{3} (1-x)^{5/2} \sqrt {1+x}+\frac {5}{2} \sin ^{-1}(x) \]
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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 = {52, 41, 222}
\begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 222
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 = 0.06, size = 56, normalized size = 0.84 \begin {gather*} \frac {\sqrt {1+x} \left (22-31 x+11 x^2-2 x^3\right )}{6 \sqrt {1-x}}+5 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 5.90, size = 133, normalized size = 1.99 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-66 \sqrt {1+x}-30 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}-17 \left (1+x\right )^{\frac {5}{2}}+2 \left (1+x\right )^{\frac {7}{2}}+59 \left (1+x\right )^{\frac {3}{2}}\right )}{6 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-59 \left (1+x\right )^{\frac {3}{2}}}{6 \sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {7}{2}}}{3 \sqrt {1-x}}+\frac {17 \left (1+x\right )^{\frac {5}{2}}}{6 \sqrt {1-x}}+5 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]+\frac {11 \sqrt {1+x}}{\sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 71, normalized size = 1.06
method | result | size |
default | \(\frac {\left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{3}+\frac {5 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{6}+\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(71\) |
risch | \(-\frac {\left (2 x^{2}-9 x +22\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 42, normalized size = 0.63 \begin {gather*} \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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 47, normalized size = 0.70 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.54, size = 173, normalized size = 2.58 \begin {gather*} \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}\: \left |{x + 1}\right | > 2 \\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 83, normalized size = 1.24 \begin {gather*} 2 \left (\left (\frac {1}{6} \sqrt {-x+1} \sqrt {-x+1}+\frac {5}{12}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {5}{4}\right ) \sqrt {-x+1} \sqrt {x+1}-5 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-x\right )}^{5/2}}{\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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