3.844 \(\int \frac{1}{\sqrt{-x} \sqrt{2-b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{-x}}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{b}} \]

[Out]

-((Sqrt[2]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[-x])/Sqrt[2]], -1])/Sqrt[b])

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Rubi [A]  time = 0.0557954, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ -\frac{\sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{-x}}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[-x])/Sqrt[2]], -1])/Sqrt[b])

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Rubi in Sympy [A]  time = 4.95835, size = 34, normalized size = 1.03 \[ - \frac{\sqrt{2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{- x}}{2} \right )}\middle | -1\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x)**(1/2)/(-b*x+2)**(1/2)/(b*x+2)**(1/2),x)

[Out]

-sqrt(2)*elliptic_f(asin(sqrt(2)*sqrt(b)*sqrt(-x)/2), -1)/sqrt(b)

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Mathematica [C]  time = 0.07164, size = 78, normalized size = 2.36 \[ \frac{2 i \sqrt{-\frac{1}{b}} b \sqrt{-x^2} \sqrt{1-\frac{4}{b^2 x^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{-\frac{1}{b}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{8-2 b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*Sqrt[-b^(-1)]*b*Sqrt[1 - 4/(b^2*x^2)]*Sqrt[-x^2]*EllipticF[I*ArcSinh[(Sqr
t[2]*Sqrt[-b^(-1)])/Sqrt[x]], -1])/Sqrt[8 - 2*b^2*x^2]

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Maple [A]  time = 0.064, size = 34, normalized size = 1. \[{\frac{1}{b}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{bx+2}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-bx}{\frac{1}{\sqrt{-x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x)^(1/2)/(-b*x+2)^(1/2)/(b*x+2)^(1/2),x)

[Out]

EllipticF(1/2*2^(1/2)*(b*x+2)^(1/2),1/2*2^(1/2))*(-b*x)^(1/2)/(-x)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{-x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(-x)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{-x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(-x)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(-x)), x)

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Sympy [A]  time = 40.2234, size = 92, normalized size = 2.79 \[ \frac{\sqrt{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{4}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac{3}{2}} \sqrt{b}} - \frac{\sqrt{2}{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x)**(1/2)/(-b*x+2)**(1/2)/(b*x+2)**(1/2),x)

[Out]

sqrt(2)*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)),
4/(b**2*x**2))/(8*pi**(3/2)*sqrt(b)) - sqrt(2)*meijerg(((-1/4, 0, 1/4, 1/2, 3/4
), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), 4*exp_polar(-2*I*pi)/(b**2*x**2))/(8*
pi**(3/2)*sqrt(b))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{-x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(-x)),x, algorithm="giac")

[Out]

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