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]

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

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ \frac {\sqrt {2 e r^2-\alpha ^2}}{2 e} \]

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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fricas [A]  time = 0.43, size = 19, normalized size = 0.83 \[ \frac {\sqrt {2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

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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giac [A]  time = 1.04, size = 19, normalized size = 0.83 \[ \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2*r^2*e - alpha^2)*e^(-1)

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maple [A]  time = 0.00, size = 20, normalized size = 0.87 \[ \frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e} \]

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 = 0.42, size = 19, normalized size = 0.83 \[ \frac {\sqrt {2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

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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mupad [B]  time = 0.29, size = 19, normalized size = 0.83 \[ \frac {\sqrt {2\,e\,r^2-\alpha ^2}}{2\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(2*e*r^2 - alpha^2)^(1/2),r)

[Out]

(2*e*r^2 - alpha^2)^(1/2)/(2*e)

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sympy [A]  time = 0.45, 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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