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

Optimal. Leaf size=81 \[ \frac{\sqrt{-\alpha ^2+2 e r^2-2 k r}}{2 e}-\frac{k \tanh ^{-1}\left (\frac{k-2 e r}{\sqrt{2} \sqrt{e} \sqrt{-\alpha ^2+2 e r^2-2 k r}}\right )}{2 \sqrt{2} e^{3/2}} \]

[Out]

Sqrt[-alpha^2 - 2*k*r + 2*e*r^2]/(2*e) - (k*ArcTanh[(k - 2*e*r)/(Sqrt[2]*Sqrt[e]
*Sqrt[-alpha^2 - 2*k*r + 2*e*r^2])])/(2*Sqrt[2]*e^(3/2))

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Rubi [A]  time = 0.0665779, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\sqrt{-\alpha ^2+2 e r^2-2 k r}}{2 e}-\frac{k \tanh ^{-1}\left (\frac{k-2 e r}{\sqrt{2} \sqrt{e} \sqrt{-\alpha ^2+2 e r^2-2 k r}}\right )}{2 \sqrt{2} e^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-alpha^2 - 2*k*r + 2*e*r^2]/(2*e) - (k*ArcTanh[(k - 2*e*r)/(Sqrt[2]*Sqrt[e]
*Sqrt[-alpha^2 - 2*k*r + 2*e*r^2])])/(2*Sqrt[2]*e^(3/2))

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Rubi in Sympy [A]  time = 4.02737, size = 75, normalized size = 0.93 \[ \frac{\sqrt{- \alpha ^{2} + 2 e r^{2} - 2 k r}}{2 e} + \frac{\sqrt{2} k \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 e r - 2 k\right )}{4 \sqrt{e} \sqrt{- \alpha ^{2} + 2 e r^{2} - 2 k r}} \right )}}{4 e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-alpha**2 + 2*e*r**2 - 2*k*r)/(2*e) + sqrt(2)*k*atanh(sqrt(2)*(4*e*r - 2*k)
/(4*sqrt(e)*sqrt(-alpha**2 + 2*e*r**2 - 2*k*r)))/(4*e**(3/2))

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Mathematica [A]  time = 0.113445, size = 84, normalized size = 1.04 \[ \frac{\sqrt{2} k \log \left (\sqrt{2} \sqrt{e} \sqrt{-\alpha ^2+2 e r^2-2 k r}+2 e r-k\right )+2 \sqrt{e} \sqrt{2 r (e r-k)-\alpha ^2}}{4 e^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e]*Sqrt[-alpha^2 + 2*r*(-k + e*r)] + Sqrt[2]*k*Log[-k + 2*e*r + Sqrt[2]*
Sqrt[e]*Sqrt[-alpha^2 - 2*k*r + 2*e*r^2]])/(4*e^(3/2))

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Maple [A]  time = 0.004, size = 70, normalized size = 0.9 \[{\frac{1}{2\,e}\sqrt{2\,e{r}^{2}-{\alpha }^{2}-2\,kr}}+{\frac{k\sqrt{2}}{4}\ln \left ({\frac{ \left ( 2\,er-k \right ) \sqrt{2}}{2}{\frac{1}{\sqrt{e}}}}+\sqrt{2\,e{r}^{2}-{\alpha }^{2}-2\,kr} \right ){e}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*e*r^2-alpha^2-2*k*r)^(1/2)/e+1/4*k/e^(3/2)*ln(1/2*(2*e*r-k)*2^(1/2)/e^(1/
2)+(2*e*r^2-alpha^2-2*k*r)^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233027, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2}{\left (k \log \left (\sqrt{2}{\left (8 \, e^{2} r^{2} - 2 \, \alpha ^{2} e - 8 \, e k r + k^{2}\right )} \sqrt{e} + 4 \,{\left (2 \, e^{2} r - e k\right )} \sqrt{2 \, e r^{2} - \alpha ^{2} - 2 \, k r}\right ) + 2 \, \sqrt{2} \sqrt{2 \, e r^{2} - \alpha ^{2} - 2 \, k r} \sqrt{e}\right )}}{8 \, e^{\frac{3}{2}}}, \frac{\sqrt{2}{\left (k \arctan \left (\frac{\sqrt{2}{\left (2 \, e r - k\right )} \sqrt{-e}}{2 \, \sqrt{2 \, e r^{2} - \alpha ^{2} - 2 \, k r} e}\right ) + \sqrt{2} \sqrt{2 \, e r^{2} - \alpha ^{2} - 2 \, k r} \sqrt{-e}\right )}}{4 \, \sqrt{-e} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*sqrt(2)*(k*log(sqrt(2)*(8*e^2*r^2 - 2*alpha^2*e - 8*e*k*r + k^2)*sqrt(e) +
4*(2*e^2*r - e*k)*sqrt(2*e*r^2 - alpha^2 - 2*k*r)) + 2*sqrt(2)*sqrt(2*e*r^2 - al
pha^2 - 2*k*r)*sqrt(e))/e^(3/2), 1/4*sqrt(2)*(k*arctan(1/2*sqrt(2)*(2*e*r - k)*s
qrt(-e)/(sqrt(2*e*r^2 - alpha^2 - 2*k*r)*e)) + sqrt(2)*sqrt(2*e*r^2 - alpha^2 -
2*k*r)*sqrt(-e))/(sqrt(-e)*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{r}{\sqrt{- \alpha ^{2} + 2 e r^{2} - 2 k r}}\, dr \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(r/sqrt(-alpha**2 + 2*e*r**2 - 2*k*r), r)

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GIAC/XCAS [A]  time = 0.258274, size = 97, normalized size = 1.2 \[ -\frac{1}{4} \, \sqrt{2} k e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{2}{\left (\sqrt{2} r e^{\frac{1}{2}} - \sqrt{2 \, r^{2} e - \alpha ^{2} - 2 \, k r}\right )} e^{\frac{1}{2}} + k \right |}\right ) + \frac{1}{2} \, \sqrt{2 \, r^{2} e - \alpha ^{2} - 2 \, k r} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*k*e^(-3/2)*ln(abs(-sqrt(2)*(sqrt(2)*r*e^(1/2) - sqrt(2*r^2*e - alph
a^2 - 2*k*r))*e^(1/2) + k)) + 1/2*sqrt(2*r^2*e - alpha^2 - 2*k*r)*e^(-1)