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}
\begin {gather*} \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \end {gather*}
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 \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 24, normalized size = 1.00
method | result | size |
default | \(\frac {r x}{\sqrt {-a^{2}-e^{2}+2 \,{\mathrm e} r^{2}}}\) | \(24\) |
norman | \(\frac {r x}{\sqrt {-a^{2}-e^{2}+2 \,{\mathrm e} r^{2}}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.57, size = 22, normalized size = 0.92 \begin {gather*} \frac {r x}{\sqrt {2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.28, size = 22, normalized size = 0.92 \begin {gather*} \frac {r x}{\sqrt {2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 20, normalized size = 0.83 \begin {gather*} \frac {r x}{\sqrt {- a^{2} - e^{2} + 2 e r^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 22, normalized size = 0.92 \begin {gather*} \frac {r x}{\sqrt {2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}
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 \begin {gather*} \frac {r\,x}{\sqrt {-a^2-e^2+2\,\mathrm {e}\,r^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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