Optimal. Leaf size=23 \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]
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Rubi [A] time = 0.0041856, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {261} \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{r}{\sqrt{-\alpha ^2+2 e r^2}} \, dr &=\frac{\sqrt{-\alpha ^2+2 e r^2}}{2 e}\\ \end{align*}
Mathematica [A] time = 0.004312, size = 23, normalized size = 1. \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 0.9 \begin{align*}{\frac{1}{2\,e}\sqrt{2\,e{r}^{2}-{\alpha }^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947773, size = 26, normalized size = 1.13 \begin{align*} \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64869, size = 42, normalized size = 1.83 \begin{align*} \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.438046, size = 29, normalized size = 1.26 \begin{align*} \begin{cases} \frac{\sqrt{- \alpha ^{2} + 2 e r^{2}}}{2 e} & \text{for}\: e \neq 0 \\\frac{r^{2}}{2 \sqrt{- \alpha ^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0891, size = 26, normalized size = 1.13 \begin{align*} \frac{1}{2} \, \sqrt{2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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