Optimal. Leaf size=32 \[ \frac {x}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {8} \[ \frac {x}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}} \, dx &=\frac {x}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.00 \[ \frac {x}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 52, normalized size = 1.62 \[ -\frac {\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} x}{2 \, K r^{5} - 2 \, H r^{3} + {\left (a^{2} + e^{2}\right )} r} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 29, normalized size = 0.91 \[ \frac {x}{\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.97 \[ \frac {x}{\sqrt {-2 K \,r^{4}+2 H \,r^{2}-a^{2}-e^{2}}\, r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 30, normalized size = 0.94 \[ \frac {x}{\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} r} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 30, normalized size = 0.94 \[ \frac {x}{r\,\sqrt {-a^2-e^2-2\,K\,r^4+2\,H\,r^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 26, normalized size = 0.81 \[ \frac {x}{r \sqrt {2 H r^{2} - 2 K r^{4} - a^{2} - e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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