Optimal. Leaf size=23 \[ \frac {\sqrt {-\alpha ^2+2 e r^2}}{2 e} \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {267}
\begin {gather*} \frac {\sqrt {2 e r^2-\alpha ^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
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*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-\alpha ^2+2 e r^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 2.13, size = 37, normalized size = 1.61 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {-\text {alpha}^2+2 e r^2}}{2 e},e\text {!=}0\right \}\right \},\frac {r^2}{2 \sqrt {-\text {alpha}^2}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 20, normalized size = 0.87
method | result | size |
gosper | \(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\) | \(20\) |
derivativedivides | \(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\) | \(20\) |
default | \(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\) | \(20\) |
trager | \(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\) | \(20\) |
risch | \(-\frac {-2 e \,r^{2}+\alpha ^{2}}{2 e \sqrt {2 e \,r^{2}-\alpha ^{2}}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 19, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 19, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 29, normalized size = 1.26 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} \frac {\sqrt {-\alpha ^{2}+2 r^{2} \mathrm {e}}}{2 \mathrm {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 19, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2\,e\,r^2-\alpha ^2}}{2\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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