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 = 0.02, 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}
\begin {gather*} \frac {x}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}} \end {gather*}
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 \begin {gather*} \frac {x}{r \sqrt {-a^2-e^2+2 H r^2-2 K r^4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.84, size = 30, normalized size = 0.94 \begin {gather*} \frac {x}{r \sqrt {2 H r^2-2 K r^4-a^2-e^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 31, normalized size = 0.97
method | result | size |
default | \(\frac {x}{r \sqrt {-2 K \,r^{4}+2 H \,r^{2}-a^{2}-e^{2}}}\) | \(31\) |
norman | \(\frac {x}{r \sqrt {-2 K \,r^{4}+2 H \,r^{2}-a^{2}-e^{2}}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 29, normalized size = 0.91 \begin {gather*} \frac {x}{\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} r} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 51, normalized size = 1.59 \begin {gather*} -\frac {\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} x}{2 \, K r^{5} - 2 \, H r^{3} + a^{2} r + r e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 26, normalized size = 0.81 \begin {gather*} \frac {x}{r \sqrt {2 H r^{2} - 2 K r^{4} - a^{2} - e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 30, normalized size = 0.94 \begin {gather*} \frac {x}{r \sqrt {2 H r^{2}-2 K r^{4}-a^{2}-\mathrm {e}^{2}}} \end {gather*}
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 \begin {gather*} \frac {x}{r\,\sqrt {-a^2-e^2-2\,K\,r^4+2\,H\,r^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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