3.209 \(\int \frac{r}{\sqrt{-\alpha ^2+2 e r^2}} \, dr\)

Optimal. Leaf size=23 \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]

[Out]

Sqrt[-alpha^2 + 2*e*r^2]/(2*e)

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Rubi [A]  time = 0.00983596, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[r/Sqrt[-alpha^2 + 2*e*r^2],r]

[Out]

Sqrt[-alpha^2 + 2*e*r^2]/(2*e)

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Rubi in Sympy [A]  time = 1.39683, size = 15, normalized size = 0.65 \[ \frac{\sqrt{- \alpha ^{2} + 2 e r^{2}}}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(r/(2*e*r**2-alpha**2)**(1/2),r)

[Out]

sqrt(-alpha**2 + 2*e*r**2)/(2*e)

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Mathematica [A]  time = 0.00633438, size = 23, normalized size = 1. \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]

Antiderivative was successfully verified.

[In]  Integrate[r/Sqrt[-alpha^2 + 2*e*r^2],r]

[Out]

Sqrt[-alpha^2 + 2*e*r^2]/(2*e)

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Maple [A]  time = 0.002, size = 20, normalized size = 0.9 \[{\frac{1}{2\,e}\sqrt{2\,e{r}^{2}-{\alpha }^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(r/(2*e*r^2-alpha^2)^(1/2),r)

[Out]

1/2*(2*e*r^2-alpha^2)^(1/2)/e

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Maxima [A]  time = 1.316, size = 26, normalized size = 1.13 \[ \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/sqrt(2*e*r^2 - alpha^2),r, algorithm="maxima")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

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Fricas [A]  time = 0.2078, size = 26, normalized size = 1.13 \[ \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/sqrt(2*e*r^2 - alpha^2),r, algorithm="fricas")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

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Sympy [A]  time = 0.764982, size = 29, normalized size = 1.26 \[ \begin{cases} \frac{\sqrt{- \alpha ^{2} + 2 e r^{2}}}{2 e} & \text{for}\: e \neq 0 \\\frac{r^{2}}{2 \sqrt{- \alpha ^{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/(2*e*r**2-alpha**2)**(1/2),r)

[Out]

Piecewise((sqrt(-alpha**2 + 2*e*r**2)/(2*e), Ne(e, 0)), (r**2/(2*sqrt(-alpha**2)
), True))

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GIAC/XCAS [A]  time = 0.200474, size = 26, normalized size = 1.13 \[ \frac{1}{2} \, \sqrt{2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/sqrt(2*e*r^2 - alpha^2),r, algorithm="giac")

[Out]

1/2*sqrt(2*r^2*e - alpha^2)*e^(-1)