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]

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

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Rubi [A]  time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {640, 621, 206} \[ \frac {\sqrt {-\alpha ^2+2 e r^2-2 k r}}{2 e}-\frac {k \tanh ^{-1}\left (\frac {k-2 e r}{\sqrt {2} \sqrt {e} \sqrt {-\alpha ^2+2 e r^2-2 k r}}\right )}{2 \sqrt {2} e^{3/2}} \]

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))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 e r^2}} \, dr &=\frac {\sqrt {-\alpha ^2-2 k r+2 e r^2}}{2 e}+\frac {k \int \frac {1}{\sqrt {-\alpha ^2-2 k r+2 e r^2}} \, dr}{2 e}\\ &=\frac {\sqrt {-\alpha ^2-2 k r+2 e r^2}}{2 e}+\frac {k \operatorname {Subst}\left (\int \frac {1}{8 e-r^2} \, dr,r,\frac {-2 k+4 e r}{\sqrt {-\alpha ^2-2 k r+2 e r^2}}\right )}{e}\\ &=\frac {\sqrt {-\alpha ^2-2 k r+2 e r^2}}{2 e}-\frac {k \tanh ^{-1}\left (\frac {k-2 e r}{\sqrt {2} \sqrt {e} \sqrt {-\alpha ^2-2 k r+2 e r^2}}\right )}{2 \sqrt {2} e^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 82, normalized size = 1.01 \[ \frac {1}{4} \left (\frac {\sqrt {2} k \tanh ^{-1}\left (\frac {2 e r-k}{\sqrt {2} \sqrt {e} \sqrt {2 r (e r-k)-\alpha ^2}}\right )}{e^{3/2}}+\frac {2 \sqrt {2 r (e r-k)-\alpha ^2}}{e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 190, normalized size = 2.35 \[ \left [\frac {\sqrt {2} \sqrt {e} k \log \left (8 \, e^{2} r^{2} - 2 \, \alpha ^{2} e - 8 \, e k r + 2 \, \sqrt {2} \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} {\left (2 \, e r - k\right )} \sqrt {e} + k^{2}\right ) + 4 \, \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} e}{8 \, e^{2}}, -\frac {\sqrt {2} \sqrt {-e} k \arctan \left (\frac {\sqrt {2} \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} {\left (2 \, e r - k\right )} \sqrt {-e}}{2 \, {\left (2 \, e^{2} r^{2} - \alpha ^{2} e - 2 \, e k r\right )}}\right ) - 2 \, \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} e}{4 \, e^{2}}\right ] \]

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, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.34, size = 72, normalized size = 0.89 \[ -\frac {1}{4} \, \sqrt {2} k e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {2} {\left (\sqrt {2} r e^{\frac {1}{2}} - \sqrt {2 \, r^{2} e - \alpha ^{2} - 2 \, k r}\right )} e^{\frac {1}{2}} + k \right |}\right ) + \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - 2 \, k r} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 70, normalized size = 0.86 \[ \frac {\sqrt {2}\, k \ln \left (\frac {\left (2 e r -k \right ) \sqrt {2}}{2 \sqrt {e}}+\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\right )}{4 e^{\frac {3}{2}}}+\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}}{2 e} \]

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 [A]  time = 0.42, size = 68, normalized size = 0.84 \[ \frac {\sqrt {2} k \log \left (4 \, e r + 2 \, \sqrt {2} \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} \sqrt {e} - 2 \, k\right )}{4 \, e^{\frac {3}{2}}} + \frac {\sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r}}{2 \, e} \]

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, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.24, size = 67, normalized size = 0.83 \[ \frac {\sqrt {-\alpha ^2+2\,e\,r^2-2\,k\,r}}{2\,e}+\frac {\sqrt {2}\,k\,\ln \left (\sqrt {-\alpha ^2+2\,e\,r^2-2\,k\,r}-\frac {\sqrt {2}\,\left (k-2\,e\,r\right )}{2\,\sqrt {e}}\right )}{4\,e^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {r}{\sqrt {- \alpha ^{2} + 2 e r^{2} - 2 k r}}\, dr \]

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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