3.2886 \(\int \frac{1}{\sqrt{4-x} \sqrt{(5-x) (-3+x)}} \, dx\)

Optimal. Leaf size=14 \[ -2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{4-x}\right ),-1\right ) \]

[Out]

-2*EllipticF[ArcSin[Sqrt[4 - x]], -1]

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Rubi [A]  time = 0.0303149, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1982, 689, 221} \[ -2 F\left (\left .\sin ^{-1}\left (\sqrt{4-x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[4 - x]*Sqrt[(5 - x)*(-3 + x)]),x]

[Out]

-2*EllipticF[ArcSin[Sqrt[4 - x]], -1]

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 689

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{4-x} \sqrt{(5-x) (-3+x)}} \, dx &=\int \frac{1}{\sqrt{4-x} \sqrt{-15+8 x-x^2}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4}} \, dx,x,\sqrt{4-x}\right )\right )\\ &=-2 F\left (\left .\sin ^{-1}\left (\sqrt{4-x}\right )\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0078893, size = 28, normalized size = 2. \[ -2 \sqrt{4-x} \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},(x-4)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[4 - x]*Sqrt[(5 - x)*(-3 + x)]),x]

[Out]

-2*Sqrt[4 - x]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, (-4 + x)^2]

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Maple [B]  time = 0.011, size = 35, normalized size = 2.5 \begin{align*} -2\,{\frac{{\it EllipticF} \left ( \sqrt{-x+4},i \right ) \sqrt{5-x}\sqrt{-3+x}}{\sqrt{- \left ( -5+x \right ) \left ( -3+x \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+4)^(1/2)/((5-x)*(-3+x))^(1/2),x)

[Out]

-2*EllipticF((-x+4)^(1/2),I)*(5-x)^(1/2)*(-3+x)^(1/2)/(-(-5+x)*(-3+x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-{\left (x - 3\right )}{\left (x - 5\right )}} \sqrt{-x + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)^(1/2)/((5-x)*(-3+x))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-(x - 3)*(x - 5))*sqrt(-x + 4)), 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 - 15} \sqrt{-x + 4}}{x^{3} - 12 \, x^{2} + 47 \, x - 60}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)^(1/2)/((5-x)*(-3+x))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^2 + 8*x - 15)*sqrt(-x + 4)/(x^3 - 12*x^2 + 47*x - 60), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 5\right ) \left (x - 3\right )} \sqrt{4 - x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)**(1/2)/((5-x)*(-3+x))**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 5)*(x - 3))*sqrt(4 - x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-{\left (x - 3\right )}{\left (x - 5\right )}} \sqrt{-x + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)^(1/2)/((5-x)*(-3+x))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-(x - 3)*(x - 5))*sqrt(-x + 4)), x)