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)])

_______________________________________________________________________________________

Rubi [A]  time = 0.059488, 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 \[ \frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}} \]

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)])

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0000575969, 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])

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 29, normalized size = 1. \[{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-2\,Kr-{a}^{2}-{e}^{2}}}}} \]

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]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.36383, size = 38, normalized size = 1.31 \[ \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r} r} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.202852, size = 38, normalized size = 1.31 \[ \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r} r} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.040436, size = 24, normalized size = 0.83 \[ \frac{x}{r \sqrt{2 H r^{2} - 2 K r - a^{2} - e^{2}}} \]

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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208267, size = 36, normalized size = 1.24 \[ \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}} r} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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