Optimal. Leaf size=30 \[ \frac{r x}{\sqrt{-a^2-e^2-2 K r^4+2 e r^2}} \]
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Rubi [A] time = 0.0153597, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {8} \[ \frac{r x}{\sqrt{-a^2-e^2-2 K r^4+2 e r^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin{align*} \int \frac{r}{\sqrt{-a^2-e^2+2 e r^2-2 K r^4}} \, dx &=\frac{r x}{\sqrt{-a^2-e^2+2 e r^2-2 K r^4}}\\ \end{align*}
Mathematica [A] time = 0.0000365, size = 30, normalized size = 1. \[ \frac{r x}{\sqrt{-a^2-e^2-2 K r^4+2 e r^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 29, normalized size = 1. \begin{align*}{rx{\frac{1}{\sqrt{-2\,K{r}^{4}+2\,E{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 = 0.929396, size = 38, normalized size = 1.27 \begin{align*} \frac{r x}{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2} - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70583, size = 101, normalized size = 3.37 \begin{align*} -\frac{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2} - e^{2}} r x}{2 \, K r^{4} - 2 \, E r^{2} + a^{2} + e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.054855, size = 26, normalized size = 0.87 \begin{align*} \frac{r x}{\sqrt{- 2 K r^{4} - a^{2} - e^{2} + 2 e r^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06962, size = 36, normalized size = 1.2 \begin{align*} \frac{r x}{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2} - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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