Optimal. Leaf size=16 \[ \frac{x}{a \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0019738, 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} \[ \frac{x}{a \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx &=\frac{x}{a \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0041616, size = 16, normalized size = 1. \[ \frac{x}{a \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 15, normalized size = 0.9 \begin{align*}{\frac{x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05934, size = 19, normalized size = 1.19 \begin{align*} \frac{x}{\sqrt{c x^{2} + a} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13287, size = 47, normalized size = 2.94 \begin{align*} \frac{\sqrt{c x^{2} + a} x}{a c x^{2} + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.848895, size = 17, normalized size = 1.06 \begin{align*} \frac{x}{a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23389, size = 19, normalized size = 1.19 \begin{align*} \frac{x}{\sqrt{c x^{2} + a} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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