Optimal. Leaf size=27 \[ \frac{r x}{\sqrt{-a^2-e^2-2 r (K-H r)}} \]
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Rubi [A] time = 0.0056563, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {8} \[ \frac{r x}{\sqrt{-a^2-e^2-2 r (K-H r)}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin{align*} \int \frac{r}{\sqrt{-a^2-e^2-2 K r+2 H r^2}} \, dx &=\frac{r x}{\sqrt{-a^2-e^2-2 r (K-H r)}}\\ \end{align*}
Mathematica [A] time = 0.0000306, size = 28, normalized size = 1.04 \[ \frac{r x}{\sqrt{-a^2-e^2+2 H r^2-2 K r}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 27, normalized size = 1. \begin{align*}{rx{\frac{1}{\sqrt{2\,H{r}^{2}-2\,Kr-{a}^{2}-{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925849, size = 35, normalized size = 1.3 \begin{align*} \frac{r x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81226, size = 53, normalized size = 1.96 \begin{align*} \frac{r x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.05365, size = 24, normalized size = 0.89 \begin{align*} \frac{r x}{\sqrt{2 H r^{2} - 2 K r - a^{2} - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05683, size = 34, normalized size = 1.26 \begin{align*} \frac{r x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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