3.29.43 \(\int \frac {1}{\sqrt {2-x} \sqrt {1+x} \sqrt {3+x}} \, dx\) [2843]

Optimal. Leaf size=24 \[ \sqrt {2} F\left (\sin ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {3}}\right )|-\frac {3}{2}\right ) \]

[Out]

EllipticF(1/3*(1+x)^(1/2)*3^(1/2),1/2*I*6^(1/2))*2^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {120} \begin {gather*} \sqrt {2} F\left (\text {ArcSin}\left (\frac {\sqrt {x+1}}{\sqrt {3}}\right )|-\frac {3}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2]*EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], -3/2]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(24)=48\).
time = 30.08, size = 67, normalized size = 2.79 \begin {gather*} -\frac {2 (3+x) \sqrt {1-\frac {5}{3+x}} \sqrt {1-\frac {2}{3+x}} F\left (\sin ^{-1}\left (\frac {\sqrt {5}}{\sqrt {3+x}}\right )|\frac {2}{5}\right )}{\sqrt {-50+35 (3+x)-5 (3+x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + x)*Sqrt[1 - 5/(3 + x)]*Sqrt[1 - 2/(3 + x)]*EllipticF[ArcSin[Sqrt[5]/Sqrt[3 + x]], 2/5])/Sqrt[-50 + 35
*(3 + x) - 5*(3 + x)^2]

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Maple [A]
time = 0.10, size = 33, normalized size = 1.38

method	result	size
default	\(\frac {2 \EllipticF \left (\frac {\sqrt {6+2 x}}{2}, \frac {\sqrt {10}}{5}\right ) \sqrt {-1-x}\, \sqrt {5}}{5 \sqrt {1+x}}\)	\(33\)
elliptic	\(\frac {\sqrt {-\left (-2+x \right ) \left (1+x \right ) \left (3+x \right )}\, \sqrt {6+2 x}\, \sqrt {10-5 x}\, \sqrt {-2-2 x}\, \EllipticF \left (\frac {\sqrt {6+2 x}}{2}, \frac {\sqrt {10}}{5}\right )}{5 \sqrt {2-x}\, \sqrt {1+x}\, \sqrt {3+x}\, \sqrt {-x^{3}-2 x^{2}+5 x +6}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*EllipticF(1/2*(6+2*x)^(1/2),1/5*10^(1/2))*(-1-x)^(1/2)*5^(1/2)/(1+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/(2-x)^(1/2)/(1+x)^(1/2)/(3+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x + 3)*sqrt(x + 1)*sqrt(-x + 2)), 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/(2-x)^(1/2)/(1+x)^(1/2)/(3+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 {2 - x} \sqrt {x + 1} \sqrt {x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {x+1}\,\sqrt {2-x}\,\sqrt {x+3}} \,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)*(x + 3)^(1/2)),x)

[Out]

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

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