Optimal. Leaf size=41 \[ -\frac {\sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 41, 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, 37}
\begin {gather*} -\frac {\sqrt {1-x}}{3 \sqrt {x+1}}-\frac {\sqrt {1-x}}{3 (x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx &=-\frac {\sqrt {1-x}}{3 (1+x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=-\frac {\sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {1-x} (2+x)}{3 (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 = 3.42, size = 68, normalized size = 1.66 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-2-x\right ) \sqrt {\frac {1-x}{1+x}}}{3 \left (1+x\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I \sqrt {1-\frac {2}{1+x}}}{3 \left (1+x\right )}-\frac {I \sqrt {1-\frac {2}{1+x}}}{3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 30, normalized size = 0.73
method | result | size |
gosper | \(-\frac {\left (2+x \right ) \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}\) | \(18\) |
default | \(-\frac {\sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}}\) | \(30\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}+x -2\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 38, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 38, normalized size = 0.93 \begin {gather*} -\frac {2 \, x^{2} + {\left (x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x + 2}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.37, size = 66, normalized size = 1.61 \begin {gather*} \begin {cases} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \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 = 47, normalized size = 1.15 \begin {gather*} -\frac {4 \left (\frac 1{4}-\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 33, normalized size = 0.80 \begin {gather*} -\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}}{\left (3\,x+3\right )\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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