3.3.12 \(\int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 e r^2}} \, dr\) [212]

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

[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 = 0.02, 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 = {654, 635, 212} \begin {gather*} \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}} \end {gather*}

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 212

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

Rule 635

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 654

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 \text {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 = 0.20, size = 86, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt {e} \sqrt {-\alpha ^2+2 r (-k+e r)}-\sqrt {2} k \log \left (-e \left (k-2 e r+\sqrt {2} \sqrt {e} \sqrt {-\alpha ^2-2 k r+2 e r^2}\right )\right )}{4 e^{3/2}} \end {gather*}

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[-(e*(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 = 0.12, size = 70, normalized size = 0.86

method	result	size
default	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}}{2 e}+\frac {k \ln \left (\frac {\left (2 e r -k \right ) \sqrt {2}}{2 \sqrt {e}}+\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\right ) \sqrt {2}}{4 e^{\frac {3}{2}}}\)	\(70\)
risch	\(-\frac {-2 e \,r^{2}+\alpha ^{2}+2 k r}{2 e \sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}}+\frac {k \ln \left (\frac {\left (2 e r -k \right ) \sqrt {2}}{2 \sqrt {e}}+\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\right ) \sqrt {2}}{4 e^{\frac {3}{2}}}\)	\(84\)

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,method=_RETURNVERBOSE)

[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 = 1.11, size = 68, normalized size = 0.84 \begin {gather*} \frac {1}{4} \, \sqrt {2} k e^{\left (-\frac {3}{2}\right )} \log \left (4 \, r e + 2 \, \sqrt {2} \sqrt {2 \, r^{2} e - \alpha ^{2} - 2 \, k r} e^{\frac {1}{2}} - 2 \, k\right ) + \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - 2 \, k r} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.56, size = 94, normalized size = 1.16 \begin {gather*} \frac {1}{8} \, {\left (\sqrt {2} k e^{\frac {1}{2}} \log \left (8 \, r^{2} e^{2} + 2 \, \sqrt {2} \sqrt {2 \, r^{2} e - \alpha ^{2} - 2 \, k r} {\left (2 \, r e - k\right )} e^{\frac {1}{2}} + k^{2} - 2 \, {\left (\alpha ^{2} + 4 \, k r\right )} e\right ) + 4 \, \sqrt {2 \, r^{2} e - \alpha ^{2} - 2 \, k r} e\right )} e^{\left (-2\right )} \end {gather*}

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)*k*e^(1/2)*log(8*r^2*e^2 + 2*sqrt(2)*sqrt(2*r^2*e - alpha^2 - 2*k*r)*(2*r*e - k)*e^(1/2) + k^2 - 2
*(alpha^2 + 4*k*r)*e) + 4*sqrt(2*r^2*e - alpha^2 - 2*k*r)*e)*e^(-2)

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

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 [A]
time = 0.50, size = 72, normalized size = 0.89 \begin {gather*} -\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 )} \end {gather*}

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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Mupad [B]
time = 0.24, size = 67, normalized size = 0.83 \begin {gather*} \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}} \end {gather*}

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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