3.2889 \(\int \frac{1}{\sqrt{(6-x) (-2+x)} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right ),\frac{4}{5}\right )}{\sqrt{5}} \]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

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Rubi [A]  time = 0.0346238, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1982, 718, 419} \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[(6 - x)*(-2 + x)]*Sqrt[-1 + x]),x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

Rule 1982

Int[(u_)^(m_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^p, x] /; FreeQ[{m, p}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 718

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

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{(6-x) (-2+x)} \sqrt{-1+x}} \, dx &=\int \frac{1}{\sqrt{-1+x} \sqrt{-12+8 x-x^2}} \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1-\frac{4 x^2}{5}}} \, dx,x,\frac{\sqrt{12-2 x}}{2 \sqrt{2}}\right )}{\sqrt{5}}\\ &=-\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [C]  time = 0.0175751, size = 74, normalized size = 2.96 \[ \frac{i \sqrt{\frac{4}{x-6}+1} \sqrt{\frac{5}{x-6}+1} (x-6)^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-6}}\right ),\frac{5}{4}\right )}{\sqrt{-(x-6) (x-2)} \sqrt{x-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[(6 - x)*(-2 + x)]*Sqrt[-1 + x]),x]

[Out]

(I*Sqrt[1 + 4/(-6 + x)]*Sqrt[1 + 5/(-6 + x)]*(-6 + x)^(3/2)*EllipticF[I*ArcSinh[2/Sqrt[-6 + x]], 5/4])/(Sqrt[-
((-6 + x)*(-2 + x))]*Sqrt[-1 + x])

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Maple [B]  time = 0.012, size = 43, normalized size = 1.7 \begin{align*} -{\frac{2\,\sqrt{5}}{5}\sqrt{-2+x}\sqrt{6-x}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ){\frac{1}{\sqrt{- \left ( x-6 \right ) \left ( -2+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((6-x)*(-2+x))^(1/2)/(-1+x)^(1/2),x)

[Out]

-2/5*(-2+x)^(1/2)*(6-x)^(1/2)*5^(1/2)*EllipticF(1/2*(6-x)^(1/2),2/5*5^(1/2))/(-(x-6)*(-2+x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-{\left (x - 2\right )}{\left (x - 6\right )}} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((6-x)*(-2+x))^(1/2)/(-1+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}{x^{3} - 9 \, x^{2} + 20 \, x - 12}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((6-x)*(-2+x))^(1/2)/(-1+x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)/(x^3 - 9*x^2 + 20*x - 12), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 6\right ) \left (x - 2\right )} \sqrt{x - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((6-x)*(-2+x))**(1/2)/(-1+x)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 6)*(x - 2))*sqrt(x - 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-{\left (x - 2\right )}{\left (x - 6\right )}} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((6-x)*(-2+x))^(1/2)/(-1+x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)), x)