Optimal. Leaf size=28 \[ \frac {\sqrt {-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {261} \[ \frac {\sqrt {-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int \frac {r}{\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}} \, dr &=\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {\sqrt {-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 24, normalized size = 0.86 \[ \frac {\sqrt {2 \, e r^{2} - \alpha ^{2} - \epsilon ^{2}}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 16, normalized size = 0.57 \[ 0.183939720586000 \, \sqrt {-1.00000000000000 \, \alpha ^{2} + 5.43656365692000 \, r^{2} - 1.00000000000000 \times 10^{-24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.89 \[ \frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 24, normalized size = 0.86 \[ \frac {\sqrt {2 \, e r^{2} - \alpha ^{2} - \epsilon ^{2}}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 24, normalized size = 0.86 \[ \frac {\sqrt {-\alpha ^2-\epsilon ^2+2\,e\,r^2}}{2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 36, normalized size = 1.29 \[ \begin {cases} \frac {\sqrt {- \alpha ^{2} + 2 e r^{2} - \epsilon ^{2}}}{2 e} & \text {for}\: e \neq 0 \\\frac {r^{2}}{2 \sqrt {- \alpha ^{2} - \epsilon ^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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