3.3.10 \(\int \frac {r}{\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}} \, dr\) [210]

Optimal. Leaf size=28 \[ \frac {\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {267} \begin {gather*} \frac {\sqrt {-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-alpha^2 - epsilon^2 + 2*e*r^2]/(2*e)

Rule 267

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

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-alpha^2 - epsilon^2 + 2*e*r^2]/(2*e)

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Maple [A]
time = 0.04, size = 25, normalized size = 0.89

method	result	size
gosper	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
derivativedivides	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
default	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
trager	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
risch	\(-\frac {-2 e \,r^{2}+\alpha ^{2}+\epsilon ^{2}}{2 e \sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(2*e*r^2-alpha^2-epsilon^2)^(1/2),r,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.80, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - \epsilon ^{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.66, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - \epsilon ^{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.28, size = 36, normalized size = 1.29 \begin {gather*} \begin {cases} \frac {\sqrt {- \alpha ^{2} + 2 e r^{2} - \epsilon ^{2}}}{2 e} & \text {for}\: e \neq 0 \\\frac {r^{2}}{2 \sqrt {- \alpha ^{2} - \epsilon ^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(2*e*r**2-alpha**2-epsilon**2)**(1/2),r)

[Out]

Piecewise((sqrt(-alpha**2 + 2*e*r**2 - epsilon**2)/(2*e), Ne(e, 0)), (r**2/(2*sqrt(-alpha**2 - epsilon**2)), T
rue))

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Giac [A]
time = 0.55, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - \epsilon ^{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.27, size = 24, normalized size = 0.86 \begin {gather*} \frac {\sqrt {-\alpha ^2-\epsilon ^2+2\,e\,r^2}}{2\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(2*e*r^2 - alpha^2 - epsilon^2)^(1/2),r)

[Out]

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

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