Optimal. Leaf size=24 \[ \frac{x}{r \sqrt{-a^2-2 r (K-H r)}} \]
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Rubi [A] time = 0.0265996, antiderivative size = 24, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0000408, 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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Maple [A] time = 0., size = 24, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-2\,Kr-{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930996, size = 31, normalized size = 1.29 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87546, size = 80, normalized size = 3.33 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r} x}{2 \, H r^{3} - a^{2} r - 2 \, K r^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.053192, size = 20, normalized size = 0.83 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - 2 K r - a^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24152, size = 31, normalized size = 1.29 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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