3.29.44 \(\int \frac {1}{\sqrt {2-x} \sqrt {3-x} \sqrt {1+x}} \, dx\) [2844]

Optimal. Leaf size=18 \[ F\left (\sin ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {3}}\right )|\frac {3}{4}\right ) \]

[Out]

EllipticF(1/3*(1+x)^(1/2)*3^(1/2),1/2*3^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {120} \begin {gather*} F\left (\text {ArcSin}\left (\frac {\sqrt {x+1}}{\sqrt {3}}\right )|\frac {3}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 - x]*Sqrt[3 - x]*Sqrt[1 + x]),x]

[Out]

EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], 3/4]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 1.28, size = 65, normalized size = 3.61 \begin {gather*} -\frac {2 i \sqrt {1-\frac {3}{2-x}} \sqrt {1+\frac {1}{2-x}} (2-x) F\left (\left .i \sinh ^{-1}\left (\frac {1}{\sqrt {2-x}}\right )\right |-3\right )}{\sqrt {-((-3+x) (1+x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 - x]*Sqrt[3 - x]*Sqrt[1 + x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.15, size = 19, normalized size = 1.06

method	result	size
default	\(\frac {2 \sqrt {3}\, \EllipticF \left (\frac {\sqrt {1+x}}{2}, \frac {2 \sqrt {3}}{3}\right )}{3}\)	\(19\)
elliptic	\(\frac {2 \sqrt {\left (-2+x \right ) \left (-3+x \right ) \left (1+x \right )}\, \sqrt {6-3 x}\, \EllipticF \left (\frac {\sqrt {1+x}}{2}, \frac {2 \sqrt {3}}{3}\right )}{3 \sqrt {2-x}\, \sqrt {x^{3}-4 x^{2}+x +6}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*3^(1/2)*EllipticF(1/2*(1+x)^(1/2),2/3*3^(1/2))

________________________________________________________________________________________

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/(2-x)^(1/2)/(3-x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 8, normalized size = 0.44 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (\frac {52}{3}, -\frac {280}{27}, x - \frac {4}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*weierstrassPInverse(52/3, -280/27, x - 4/3)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 - x} \sqrt {3 - x} \sqrt {x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(2 - x)*sqrt(3 - x)*sqrt(x + 1)), x)

________________________________________________________________________________________

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/(2-x)^(1/2)/(3-x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{\sqrt {x+1}\,\sqrt {2-x}\,\sqrt {3-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________