3.29.89 \(\int \frac {1}{\sqrt {(6-x) (-2+x)} \sqrt {-1+x}} \, dx\) [2889]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {1976, 732, 430} \begin {gather*} -\frac {2 F\left (\text {ArcSin}\left (\frac {\sqrt {6-x}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}} \end {gather*}

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 430

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

Rule 732

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 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.02, size = 74, normalized size = 2.96 \begin {gather*} \frac {i \sqrt {1+\frac {4}{-6+x}} \sqrt {1+\frac {5}{-6+x}} (-6+x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-6+x}}\right )|\frac {5}{4}\right )}{\sqrt {-((-6+x) (-2+x))} \sqrt {-1+x}} \end {gather*}

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 [A]
time = 0.12, size = 38, normalized size = 1.52

method	result	size
default	\(\frac {2 \EllipticF \left (\sqrt {-1+x}, \frac {\sqrt {5}}{5}\right ) \sqrt {2-x}\, \sqrt {30-5 x}}{5 \sqrt {-\left (-2+x \right ) \left (-6+x \right )}}\)	\(38\)
elliptic	\(\frac {2 \sqrt {-\left (-6+x \right ) \left (-2+x \right ) \left (-1+x \right )}\, \sqrt {30-5 x}\, \sqrt {2-x}\, \EllipticF \left (\sqrt {-1+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {-\left (-2+x \right ) \left (-6+x \right )}\, \sqrt {-x^{3}+9 x^{2}-20 x +12}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*EllipticF((-1+x)^(1/2),1/5*5^(1/2))*(2-x)^(1/2)*(30-5*x)^(1/2)/(-(-2+x)*(-6+x))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (x - 6\right ) \left (x - 2\right )} \sqrt {x - 1}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {-\left (x-2\right )\,\left (x-6\right )}\,\sqrt {x-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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