Optimal. Leaf size=45 \[ -\frac {4 (x+1)}{15 \left (-4 x^2+2 x+1\right )^{3/2}}-\frac {122 x+7}{75 \sqrt {-4 x^2+2 x+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1660, 636} \[ -\frac {4 (x+1)}{15 \left (-4 x^2+2 x+1\right )^{3/2}}-\frac {122 x+7}{75 \sqrt {-4 x^2+2 x+1}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 1660
Rubi steps
\begin {align*} \int \frac {-1-8 x+8 x^3}{\left (1+2 x-4 x^2\right )^{5/2}} \, dx &=-\frac {4 (1+x)}{15 \left (1+2 x-4 x^2\right )^{3/2}}-\frac {1}{30} \int \frac {46+60 x}{\left (1+2 x-4 x^2\right )^{3/2}} \, dx\\ &=-\frac {4 (1+x)}{15 \left (1+2 x-4 x^2\right )^{3/2}}-\frac {7+122 x}{75 \sqrt {1+2 x-4 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 33, normalized size = 0.73 \[ -\frac {-488 x^3+216 x^2+156 x+27}{75 \left (-4 x^2+2 x+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 45, normalized size = 1.00 \[ \frac {\sqrt {-4 x^2+2 x+1} \left (488 x^3-216 x^2-156 x-27\right )}{75 \left (4 x^2-2 x-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 73, normalized size = 1.62 \[ -\frac {432 \, x^{4} - 432 \, x^{3} - 108 \, x^{2} - {\left (488 \, x^{3} - 216 \, x^{2} - 156 \, x - 27\right )} \sqrt {-4 \, x^{2} + 2 \, x + 1} + 108 \, x + 27}{75 \, {\left (16 \, x^{4} - 16 \, x^{3} - 4 \, x^{2} + 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 41, normalized size = 0.91 \[ \frac {{\left (4 \, {\left (2 \, {\left (61 \, x - 27\right )} x - 39\right )} x - 27\right )} \sqrt {-4 \, x^{2} + 2 \, x + 1}}{75 \, {\left (4 \, x^{2} - 2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 30, normalized size = 0.67
method | result | size |
gosper | \(\frac {488 x^{3}-216 x^{2}-156 x -27}{75 \left (-4 x^{2}+2 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
trager | \(\frac {\left (488 x^{3}-216 x^{2}-156 x -27\right ) \sqrt {-4 x^{2}+2 x +1}}{75 \left (4 x^{2}-2 x -1\right )^{2}}\) | \(42\) |
risch | \(-\frac {488 x^{3}-216 x^{2}-156 x -27}{75 \left (4 x^{2}-2 x -1\right ) \sqrt {-4 x^{2}+2 x +1}}\) | \(42\) |
default | \(\frac {2 x^{2}}{\left (-4 x^{2}+2 x +1\right )^{\frac {3}{2}}}-\frac {x}{4 \left (-4 x^{2}+2 x +1\right )^{\frac {3}{2}}}-\frac {49}{48 \left (-4 x^{2}+2 x +1\right )^{\frac {3}{2}}}+\frac {\frac {61}{240}-\frac {61 x}{60}}{\left (-4 x^{2}+2 x +1\right )^{\frac {3}{2}}}+\frac {\frac {61}{150}-\frac {122 x}{75}}{\sqrt {-4 x^{2}+2 x +1}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 76, normalized size = 1.69 \[ -\frac {122 \, x}{75 \, \sqrt {-4 \, x^{2} + 2 \, x + 1}} + \frac {2 \, x^{2}}{{\left (-4 \, x^{2} + 2 \, x + 1\right )}^{\frac {3}{2}}} + \frac {61}{150 \, \sqrt {-4 \, x^{2} + 2 \, x + 1}} - \frac {19 \, x}{15 \, {\left (-4 \, x^{2} + 2 \, x + 1\right )}^{\frac {3}{2}}} - \frac {23}{30 \, {\left (-4 \, x^{2} + 2 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 29, normalized size = 0.64 \[ -\frac {-488\,x^3+216\,x^2+156\,x+27}{75\,{\left (-4\,x^2+2\,x+1\right )}^{3/2}} \]
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 \[ \int \frac {8 x^{3} - 8 x - 1}{\left (- 4 x^{2} + 2 x + 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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