Optimal. Leaf size=43 \[ \frac{2 (x+2)}{243 \sqrt{-x^2-4 x+5}}+\frac{x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]
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Rubi [A] time = 0.0075282, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {614, 613} \[ \frac{2 (x+2)}{243 \sqrt{-x^2-4 x+5}}+\frac{x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 614
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{\left (5-4 x-x^2\right )^{5/2}} \, dx &=\frac{2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac{2}{27} \int \frac{1}{\left (5-4 x-x^2\right )^{3/2}} \, dx\\ &=\frac{2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac{2 (2+x)}{243 \sqrt{5-4 x-x^2}}\\ \end{align*}
Mathematica [A] time = 0.0093429, size = 31, normalized size = 0.72 \[ -\frac{(x+2) \left (2 x^2+8 x-19\right )}{243 \left (-x^2-4 x+5\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{ \left ( 5+x \right ) \left ( -1+x \right ) \left ( 2\,{x}^{3}+12\,{x}^{2}-3\,x-38 \right ) }{243} \left ( -{x}^{2}-4\,x+5 \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.921316, size = 80, normalized size = 1.86 \begin{align*} \frac{2 \, x}{243 \, \sqrt{-x^{2} - 4 \, x + 5}} + \frac{4}{243 \, \sqrt{-x^{2} - 4 \, x + 5}} + \frac{x}{27 \,{\left (-x^{2} - 4 \, x + 5\right )}^{\frac{3}{2}}} + \frac{2}{27 \,{\left (-x^{2} - 4 \, x + 5\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30562, size = 123, normalized size = 2.86 \begin{align*} -\frac{{\left (2 \, x^{3} + 12 \, x^{2} - 3 \, x - 38\right )} \sqrt{-x^{2} - 4 \, x + 5}}{243 \,{\left (x^{4} + 8 \, x^{3} + 6 \, x^{2} - 40 \, x + 25\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- x^{2} - 4 x + 5\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07263, size = 49, normalized size = 1.14 \begin{align*} -\frac{{\left ({\left (2 \,{\left (x + 6\right )} x - 3\right )} x - 38\right )} \sqrt{-x^{2} - 4 \, x + 5}}{243 \,{\left (x^{2} + 4 \, x - 5\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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