Optimal. Leaf size=24 \[ \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, 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 = {8} \[ \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \]
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}} \, dx &=\frac {r x}{\sqrt {-a^2-e^2+2 e r^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 22, normalized size = 0.92 \[ \frac {r x}{\sqrt {2 \, E r^{2} - a^{2} - e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 21, normalized size = 0.88 \[ \frac {r x}{\sqrt {2 \, E r^{2} - a^{2} - e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.96 \[ \frac {r x}{\sqrt {2 E \,r^{2}-a^{2}-e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 22, normalized size = 0.92 \[ \frac {r x}{\sqrt {2 \, E r^{2} - a^{2} - e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 23, normalized size = 0.96 \[ \frac {r\,x}{\sqrt {-a^2-e^2+2\,\mathrm {e}\,r^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 20, normalized size = 0.83 \[ \frac {r x}{\sqrt {- a^{2} - e^{2} + 2 e r^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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