3.29.86 \(\int \frac {1}{\sqrt {4-x} \sqrt {(5-x) (-3+x)}} \, dx\) [2886]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1976, 703, 227} \begin {gather*} -2 F\left (\left .\text {ArcSin}\left (\sqrt {4-x}\right )\right |-1\right ) \end {gather*}

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 227

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]

Rule 703

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(4/e)*Sqrt[-c/(b^2
- 4*a*c)], 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 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 {4-x} \sqrt {(5-x) (-3+x)}} \, dx &=\int \frac {1}{\sqrt {4-x} \sqrt {-15+8 x-x^2}} \, dx\\ &=-\left (2 \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.01, size = 28, normalized size = 2.00 \begin {gather*} -2 \sqrt {4-x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(-4+x)^2\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
time = 0.12, size = 39, normalized size = 2.79

method	result	size
default	\(\frac {\EllipticF \left (\frac {\sqrt {-6+2 x}}{2}, \sqrt {2}\right ) \sqrt {10-2 x}\, \sqrt {-6+2 x}}{\sqrt {-\left (-5+x \right ) \left (-3+x \right )}}\)	\(39\)
elliptic	\(\frac {\sqrt {\left (-4+x \right ) \left (-5+x \right ) \left (-3+x \right )}\, \sqrt {-6+2 x}\, \sqrt {10-2 x}\, \EllipticF \left (\frac {\sqrt {-6+2 x}}{2}, \sqrt {2}\right )}{\sqrt {-\left (-5+x \right ) \left (-3+x \right )}\, \sqrt {x^{3}-12 x^{2}+47 x -60}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticF(1/2*(-6+2*x)^(1/2),2^(1/2))*(10-2*x)^(1/2)*(-6+2*x)^(1/2)/(-(-5+x)*(-3+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/(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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.20, size = 8, normalized size = 0.57 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (4, 0, x - 4\right ) \end {gather*}

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]

2*weierstrassPInverse(4, 0, x - 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (x - 5\right ) \left (x - 3\right )} \sqrt {4 - x}}\, dx \end {gather*}

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 = 0.00, 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/(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {1}{\sqrt {-\left (x-3\right )\,\left (x-5\right )}\,\sqrt {4-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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