Optimal. Leaf size=42 \[ \frac {1}{3 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{3 \sqrt {1-x} \sqrt {1+x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 39}
\begin {gather*} \frac {2 x}{3 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx &=\frac {1}{3 (1-x)^{3/2} \sqrt {1+x}}+\frac {2}{3} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{3 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{3 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 30, normalized size = 0.71 \begin {gather*} \frac {-1-2 x+2 x^2}{3 (-1+x) \sqrt {1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 5.37, size = 133, normalized size = 3.17 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (1+2 x-2 x^2\right ) \sqrt {\frac {1-x}{1+x}}}{3 \left (1-2 x+x^2\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-3 I \sqrt {1-\frac {2}{1+x}}}{-12 x+3 \left (1+x\right )^2}-\frac {2 I \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{-12 x+3 \left (1+x\right )^2}+\frac {I 6 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{-12 x+3 \left (1+x\right )^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 44, normalized size = 1.05
method | result | size |
gosper | \(-\frac {2 x^{2}-2 x -1}{3 \sqrt {1+x}\, \left (1-x \right )^{\frac {3}{2}}}\) | \(25\) |
default | \(\frac {1}{3 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {2}{3 \sqrt {1-x}\, \sqrt {1+x}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\) | \(44\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{2}-2 x -1\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 40, normalized size = 0.95 \begin {gather*} \frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} - \frac {1}{3 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 54, normalized size = 1.29 \begin {gather*} \frac {x^{3} - x^{2} - {\left (2 \, x^{2} - 2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - x + 1}{3 \, {\left (x^{3} - x^{2} - x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.47, size = 160, normalized size = 3.81 \begin {gather*} \begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac {6 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac {3 \sqrt {-1 + \frac {2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac {6 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac {3 i \sqrt {1 - \frac {2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (30) = 60\).
time = 0.01, size = 171, normalized size = 4.07 \begin {gather*} -2 \left (\frac {-\frac {4096}{3} \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}+\frac {14336 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{262144}+\frac {21 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+1}{192 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}}+\frac {\sqrt {-x+1} \sqrt {x+1}}{8 \left (x+1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 42, normalized size = 1.00 \begin {gather*} \frac {2\,x\,\sqrt {1-x}+\sqrt {1-x}-2\,x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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