Optimal. Leaf size=16 \[ -\frac{x}{25 \sqrt{4 x^2-25}} \]
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Rubi [A] time = 0.0014153, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {191} \[ -\frac{x}{25 \sqrt{4 x^2-25}} \]
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (-25+4 x^2\right )^{3/2}} \, dx &=-\frac{x}{25 \sqrt{-25+4 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0038095, size = 16, normalized size = 1. \[ -\frac{x}{25 \sqrt{4 x^2-25}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,x-5 \right ) \left ( 5+2\,x \right ) x}{25} \left ( 4\,{x}^{2}-25 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.920857, size = 16, normalized size = 1. \begin{align*} -\frac{x}{25 \, \sqrt{4 \, x^{2} - 25}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06223, size = 76, normalized size = 4.75 \begin{align*} -\frac{4 \, x^{2} + 2 \, \sqrt{4 \, x^{2} - 25} x - 25}{50 \,{\left (4 \, x^{2} - 25\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.721938, size = 36, normalized size = 2.25 \begin{align*} \begin{cases} - \frac{x}{25 \sqrt{4 x^{2} - 25}} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{25} > 1 \\\frac{i x}{25 \sqrt{25 - 4 x^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06364, size = 16, normalized size = 1. \begin{align*} -\frac{x}{25 \, \sqrt{4 \, x^{2} - 25}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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