Optimal. Leaf size=32 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}} \]
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Rubi [A] time = 0.0256325, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}} \]
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^2-2 K r^4}} \, dx &=\frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}}\\ \end{align*}
Mathematica [A] time = 0.0000345, size = 32, normalized size = 1. \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{-2\,K{r}^{4}+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 = 0.932703, size = 41, normalized size = 1.28 \begin{align*} \frac{x}{\sqrt{-2 \, K r^{4} + 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 = 1.81301, size = 104, normalized size = 3.25 \begin{align*} -\frac{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} x}{2 \, K r^{5} - 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 = 0.055048, size = 26, normalized size = 0.81 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - 2 K r^{4} - a^{2} - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04741, size = 39, normalized size = 1.22 \begin{align*} \frac{x}{\sqrt{-2 \, K r^{4} + 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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