Optimal. Leaf size=24 \[ \frac {x}{r \sqrt {-a^2-2 r (K-H r)}} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {8} \[ \frac {x}{r \sqrt {-a^2-2 r (K-H r)}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-a^2-2 K r+2 H r^2}} \, dx &=\frac {x}{r \sqrt {-a^2-2 r (K-H r)}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.04 \[ \frac {x}{r \sqrt {-a^2+2 H r^2-2 K r}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 41, normalized size = 1.71 \[ \frac {\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r} x}{2 \, H r^{3} - a^{2} r - 2 \, K r^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 23, normalized size = 0.96 \[ \frac {x}{\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r} r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {x}{\sqrt {2 H \,r^{2}-2 K r -a^{2}}\, r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 23, normalized size = 0.96 \[ \frac {x}{\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r} r} \]
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 \[ \frac {x}{r\,\sqrt {-a^2+2\,H\,r^2-2\,K\,r}} \]
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 \[ \frac {x}{r \sqrt {2 H r^{2} - 2 K r - a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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