Optimal. Leaf size=23 \[ \frac{5-2 x}{9 \sqrt{-x^2-4 x+5}} \]
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Rubi [A] time = 0.0045149, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {636} \[ \frac{5-2 x}{9 \sqrt{-x^2-4 x+5}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin{align*} \int \frac{x}{\left (5-4 x-x^2\right )^{3/2}} \, dx &=\frac{5-2 x}{9 \sqrt{5-4 x-x^2}}\\ \end{align*}
Mathematica [A] time = 0.0296655, size = 23, normalized size = 1. \[ \frac{5-2 x}{9 \sqrt{-x^2-4 x+5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 26, normalized size = 1.1 \begin{align*}{\frac{ \left ( x+5 \right ) \left ( -1+x \right ) \left ( 2\,x-5 \right ) }{9} \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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Maxima [A] time = 1.18211, size = 41, normalized size = 1.78 \begin{align*} -\frac{2 \, x}{9 \, \sqrt{-x^{2} - 4 \, x + 5}} + \frac{5}{9 \, \sqrt{-x^{2} - 4 \, x + 5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92854, size = 70, normalized size = 3.04 \begin{align*} \frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (2 \, x - 5\right )}}{9 \,{\left (x^{2} + 4 \, x - 5\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{x}{\left (- \left (x - 1\right ) \left (x + 5\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32979, size = 39, normalized size = 1.7 \begin{align*} \frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (2 \, x - 5\right )}}{9 \,{\left (x^{2} + 4 \, x - 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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