Optimal. Leaf size=26 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2}} \]
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Rubi [A] time = 0.0194002, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin{align*} \int \frac{1}{r \sqrt{-a^2-e^2+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2-e^2+2 H r^2}}\\ \end{align*}
Mathematica [A] time = 0.0000479, size = 26, normalized size = 1. \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-{a}^{2}-{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935708, size = 32, normalized size = 1.23 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2}} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78943, size = 74, normalized size = 2.85 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2} - e^{2}} x}{2 \, H r^{3} -{\left (a^{2} + e^{2}\right )} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.062354, size = 19, normalized size = 0.73 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - a^{2} - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05253, size = 31, normalized size = 1.19 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2}} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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