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

Optimal. Leaf size=32 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}} \]

[Out]

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

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Rubi [A]  time = 0.0470756, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{r}\, dx}{\sqrt{2 H r^{2} - 2 K r^{4} - a^{2} - e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/r/(-2*K*r**4+2*H*r**2-a**2-e**2)**(1/2),x)

[Out]

Integral(1/r, x)/sqrt(2*H*r**2 - 2*K*r**4 - a**2 - e**2)

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Mathematica [A]  time = 0.0000492774, size = 32, normalized size = 1. \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 31, normalized size = 1. \[{\frac{x}{r}{\frac{1}{\sqrt{-2\,K{r}^{4}+2\,H{r}^{2}-{a}^{2}-{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34838, size = 41, normalized size = 1.28 \[ \frac{x}{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} r} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.200655, size = 41, normalized size = 1.28 \[ \frac{x}{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} r} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.040091, size = 26, normalized size = 0.81 \[ \frac{x}{r \sqrt{2 H r^{2} - 2 K r^{4} - a^{2} - e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.200058, size = 39, normalized size = 1.22 \[ \frac{x}{\sqrt{-2 \, K r^{4} + 2 \, H r^{2} - a^{2} - e^{2}} r} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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