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 = 0.0041856, 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, Rules used = {261} \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]

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)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{r}{\sqrt{-\alpha ^2+2 e r^2}} \, dr &=\frac{\sqrt{-\alpha ^2+2 e r^2}}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.004312, 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 = 0.003, size = 20, normalized size = 0.9 \begin{align*}{\frac{1}{2\,e}\sqrt{2\,e{r}^{2}-{\alpha }^{2}}} \end{align*}

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 = 0.947773, size = 26, normalized size = 1.13 \begin{align*} \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(2*e*r^2-alpha^2)^(1/2),r, algorithm="maxima")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

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Fricas [A]  time = 1.64869, size = 42, normalized size = 1.83 \begin{align*} \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(2*e*r^2-alpha^2)^(1/2),r, algorithm="fricas")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

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Sympy [A]  time = 0.438046, size = 29, normalized size = 1.26 \begin{align*} \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} \end{align*}

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 [A]  time = 1.0891, size = 26, normalized size = 1.13 \begin{align*} \frac{1}{2} \, \sqrt{2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(2*e*r^2-alpha^2)^(1/2),r, algorithm="giac")

[Out]

1/2*sqrt(2*r^2*e - alpha^2)*e^(-1)