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

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

[Out]

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

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Rubi [A]  time = 0.0077165, 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, Rules used = {116} \[ -\frac{\sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{-x}}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{b}} \]

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])

Rule 116

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

Rubi steps

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

Mathematica [C]  time = 0.0083511, size = 29, normalized size = 0.88 \[ \frac{x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b^2 x^2}{4}\right )}{\sqrt{-x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Hypergeometric2F1[1/4, 1/2, 5/4, (b^2*x^2)/4])/Sqrt[-x]

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Maple [A]  time = 0.038, size = 34, normalized size = 1. \begin{align*}{\frac{1}{b}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{bx+2}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-bx}{\frac{1}{\sqrt{-x}}}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{-x}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{-x}}{b^{2} x^{3} - 4 \, x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(-x)/(b^2*x^3 - 4*x), x)

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Sympy [B]  time = 15.1182, size = 92, normalized size = 2.79 \begin{align*} \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}} \end{align*}

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)*s
qrt(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{-x}}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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