Optimal. Leaf size=25 \[ \frac{r x}{\sqrt{-a^2-2 K r^4+2 e r^2}} \]
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Rubi [A] time = 0.0127465, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {8} \[ \frac{r x}{\sqrt{-a^2-2 K r^4+2 e r^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin{align*} \int \frac{r}{\sqrt{-a^2+2 e r^2-2 K r^4}} \, dx &=\frac{r x}{\sqrt{-a^2+2 e r^2-2 K r^4}}\\ \end{align*}
Mathematica [A] time = 0.0000295, size = 25, normalized size = 1. \[ \frac{r x}{\sqrt{-a^2-2 K r^4+2 e r^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 24, normalized size = 1. \begin{align*}{rx{\frac{1}{\sqrt{-2\,K{r}^{4}+2\,E{r}^{2}-{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930996, size = 31, normalized size = 1.24 \begin{align*} \frac{r x}{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88055, size = 85, normalized size = 3.4 \begin{align*} -\frac{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2}} r x}{2 \, K r^{4} - 2 \, E r^{2} + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.053411, size = 22, normalized size = 0.88 \begin{align*} \frac{r x}{\sqrt{- 2 K r^{4} - a^{2} + 2 e r^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06028, size = 31, normalized size = 1.24 \begin{align*} \frac{r x}{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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