3.105 \(\int \frac{1}{r \sqrt{-a^2+2 H r^2-2 K r^4}} \, dx\)

Optimal. Leaf size=27 \[ \frac{x}{r \sqrt{-a^2+2 H r^2-2 K r^4}} \]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4])

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Rubi [A]  time = 0.0227884, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2+2 H r^2-2 K r^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4]),x]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4])

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{r \sqrt{-a^2+2 H r^2-2 K r^4}} \, dx &=\frac{x}{r \sqrt{-a^2+2 H r^2-2 K r^4}}\\ \end{align*}

Mathematica [A]  time = 0.0000354, size = 27, normalized size = 1. \[ \frac{x}{r \sqrt{-a^2+2 H r^2-2 K r^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4]),x]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4])

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Maple [A]  time = 0., size = 26, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{-2\,K{r}^{4}+2\,H{r}^{2}-{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x)

[Out]

x/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2)

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Maxima [A]  time = 0.927586, size = 34, normalized size = 1.26 \begin{align*} \frac{x}{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x, algorithm="maxima")

[Out]

x/(sqrt(-2*K*r^4 + 2*H*r^2 - a^2)*r)

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Fricas [A]  time = 1.8654, size = 85, normalized size = 3.15 \begin{align*} -\frac{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2}} x}{2 \, K r^{5} - 2 \, H r^{3} + a^{2} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-2*K*r^4 + 2*H*r^2 - a^2)*x/(2*K*r^5 - 2*H*r^3 + a^2*r)

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Sympy [A]  time = 0.054584, size = 22, normalized size = 0.81 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - 2 K r^{4} - a^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r**4+2*H*r**2-a**2)**(1/2),x)

[Out]

x/(r*sqrt(2*H*r**2 - 2*K*r**4 - a**2))

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Giac [A]  time = 1.05276, size = 34, normalized size = 1.26 \begin{align*} \frac{x}{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x, algorithm="giac")

[Out]

x/(sqrt(-2*K*r^4 + 2*H*r^2 - a^2)*r)