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 = 0.0216626, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 1660
Rule 636
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*}
Mathematica [A] time = 0.112204, 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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Maple [A] time = 0.005, size = 30, normalized size = 0.7 \begin{align*}{\frac{488\,{x}^{3}-216\,{x}^{2}-156\,x-27}{75} \left ( -4\,{x}^{2}+2\,x+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.955753, size = 103, normalized size = 2.29 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02849, size = 194, normalized size = 4.31 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09662, size = 55, normalized size = 1.22 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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