Optimal. Leaf size=58 \[ \frac {1}{\sqrt {1-x} (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37}
\begin {gather*} -\frac {2 \sqrt {1-x}}{3 \sqrt {x+1}}-\frac {2 \sqrt {1-x}}{3 (x+1)^{3/2}}+\frac {1}{(x+1)^{3/2} \sqrt {1-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx &=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}+2 \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx\\ &=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 (1+x)^{3/2}}+\frac {2}{3} \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 30, normalized size = 0.52 \begin {gather*} \frac {-1+2 x+2 x^2}{3 \sqrt {1-x} (1+x)^{3/2}} \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 = 5.46, size = 133, normalized size = 2.29 \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+x^2\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-2 I \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{-6-6 x+3 \left (1+x\right )^2}+\frac {I 2 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{-6-6 x+3 \left (1+x\right )^2}+\frac {I \sqrt {1-\frac {2}{1+x}}}{-6-6 x+3 \left (1+x\right )^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 43, normalized size = 0.74
method | result | size |
gosper | \(\frac {2 x^{2}+2 x -1}{3 \left (1+x \right )^{\frac {3}{2}} \sqrt {1-x}}\) | \(25\) |
default | \(\frac {1}{\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\) | \(43\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{2}+2 x -1\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 38, normalized size = 0.66 \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 = 0.30, size = 49, normalized size = 0.84 \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 [A]
time = 3.49, size = 167, normalized size = 2.88 \begin {gather*} \begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {\sqrt {-1 + \frac {2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \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}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {i \sqrt {1 - \frac {2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 111, normalized size = 1.91 \begin {gather*} 2 \left (\frac {\sqrt {-x+1}}{8 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{32 \sqrt {-x+1}}+\frac {2 \left (\frac {5}{48} \sqrt {-x+1} \sqrt {-x+1}-\frac 1{4}\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 48, normalized size = 0.83 \begin {gather*} -\frac {2\,x\,\sqrt {1-x}-\sqrt {1-x}+2\,x^2\,\sqrt {1-x}}{\left (3\,x^2-3\right )\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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