3.104 \(\int \frac{1}{r \sqrt{-a^2-e^2+2 H r^2}} \, dx\)

Optimal. Leaf size=26 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2}} \]

[Out]

x/(r*Sqrt[-a^2 - e^2 + 2*H*r^2])

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Rubi [A]  time = 0.0194002, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(r*Sqrt[-a^2 - e^2 + 2*H*r^2]),x]

[Out]

x/(r*Sqrt[-a^2 - e^2 + 2*H*r^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{r \sqrt{-a^2-e^2+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2-e^2+2 H r^2}}\\ \end{align*}

Mathematica [A]  time = 0.0000479, size = 26, normalized size = 1. \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-a^2 - e^2 + 2*H*r^2]),x]

[Out]

x/(r*Sqrt[-a^2 - e^2 + 2*H*r^2])

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Maple [A]  time = 0.002, size = 25, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-{a}^{2}-{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/r/(2*H*r^2-a^2-e^2)^(1/2),x)

[Out]

x/r/(2*H*r^2-a^2-e^2)^(1/2)

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Maxima [A]  time = 0.935708, size = 32, normalized size = 1.23 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r^2-a^2-e^2)^(1/2),x, algorithm="maxima")

[Out]

x/(sqrt(2*H*r^2 - a^2 - e^2)*r)

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Fricas [A]  time = 1.78943, size = 74, normalized size = 2.85 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2} - e^{2}} x}{2 \, H r^{3} -{\left (a^{2} + e^{2}\right )} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r^2-a^2-e^2)^(1/2),x, algorithm="fricas")

[Out]

sqrt(2*H*r^2 - a^2 - e^2)*x/(2*H*r^3 - (a^2 + e^2)*r)

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Sympy [A]  time = 0.062354, size = 19, normalized size = 0.73 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - a^{2} - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r**2-a**2-e**2)**(1/2),x)

[Out]

x/(r*sqrt(2*H*r**2 - a**2 - e**2))

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Giac [A]  time = 1.05253, size = 31, normalized size = 1.19 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r^2-a^2-e^2)^(1/2),x, algorithm="giac")

[Out]

x/(sqrt(2*H*r^2 - a^2 - e^2)*r)