Optimal. Leaf size=43 \[ \frac {1-x}{6 \left (x^2-2 x-1\right )^{3/2}}-\frac {1-x}{6 \sqrt {x^2-2 x-1}} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {614, 613} \[ \frac {1-x}{6 \left (x^2-2 x-1\right )^{3/2}}-\frac {1-x}{6 \sqrt {x^2-2 x-1}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rubi steps
\begin {align*} \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx &=\frac {1-x}{6 \left (-1-2 x+x^2\right )^{3/2}}-\frac {1}{3} \int \frac {1}{\left (-1-2 x+x^2\right )^{3/2}} \, dx\\ &=\frac {1-x}{6 \left (-1-2 x+x^2\right )^{3/2}}-\frac {1-x}{6 \sqrt {-1-2 x+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.60 \[ \frac {x^3-3 x^2+2}{6 \left (x^2-2 x-1\right )^{3/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 27, normalized size = 0.63 \[ \frac {(x-1) \left (x^2-2 x-2\right )}{6 \left (x^2-2 x-1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 61, normalized size = 1.42 \[ \frac {x^{4} - 4 \, x^{3} + 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2} + 2\right )} \sqrt {x^{2} - 2 \, x - 1} + 4 \, x + 1}{6 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 21, normalized size = 0.49 \[ \frac {{\left (x - 3\right )} x^{2} + 2}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 23, normalized size = 0.53
method | result | size |
gosper | \(\frac {x^{3}-3 x^{2}+2}{6 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}\) | \(23\) |
trager | \(\frac {x^{3}-3 x^{2}+2}{6 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}\) | \(23\) |
risch | \(\frac {x^{3}-3 x^{2}+2}{6 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}\) | \(23\) |
default | \(-\frac {2 x -2}{12 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}+\frac {2 x -2}{12 \sqrt {x^{2}-2 x -1}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 51, normalized size = 1.19 \[ \frac {x}{6 \, \sqrt {x^{2} - 2 \, x - 1}} - \frac {1}{6 \, \sqrt {x^{2} - 2 \, x - 1}} - \frac {x}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} + \frac {1}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 22, normalized size = 0.51 \[ \frac {x^3-3\,x^2+2}{6\,{\left (x^2-2\,x-1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x^{2} - 2 x - 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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