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

Optimal. Leaf size=29 \[ \frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}} \]

[Out]

x/(r*Sqrt[-a^2 - e^2 - 2*r*(K - H*r)])

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Rubi [A]  time = 0.0316036, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(r*Sqrt[-a^2 - e^2 - 2*r*(K - H*r)])

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 K r+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}}\\ \end{align*}

Mathematica [A]  time = 0.000036, size = 30, normalized size = 1.03 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 29, normalized size = 1. \begin{align*}{\frac{x}{r}{\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.

[In]

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

[Out]

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

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Maxima [A]  time = 0.928842, size = 38, normalized size = 1.31 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.81653, size = 99, normalized size = 3.41 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r} x}{2 \, H r^{3} - 2 \, K r^{2} -{\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^2-2*K*r-a^2-e^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.054356, size = 24, normalized size = 0.83 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - 2 K r - a^{2} - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(r*sqrt(2*H*r**2 - 2*K*r - a**2 - e**2))

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Giac [A]  time = 1.07776, size = 36, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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