3.853 \(\int \frac{1}{\sqrt{-x} \sqrt{a-b x} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sqrt{\frac{b x}{a}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{-x}}{\sqrt{a}}\right ),-1\right )}{\sqrt{b} \sqrt{a-b x} \sqrt{a+b x}} \]

[Out]

(-2*Sqrt[a]*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[-x])/Sqrt[a]], -1])/(Sqrt[b]*Sq
rt[a - b*x]*Sqrt[a + b*x])

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Rubi [A]  time = 0.0211199, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {117, 116} \[ -\frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sqrt{\frac{b x}{a}+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{-x}}{\sqrt{a}}\right )\right |-1\right )}{\sqrt{b} \sqrt{a-b x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a]*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[-x])/Sqrt[a]], -1])/(Sqrt[b]*Sq
rt[a - b*x]*Sqrt[a + b*x])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

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

Mathematica [C]  time = 0.0241342, size = 66, normalized size = 0.86 \[ \frac{2 x \sqrt{1-\frac{b^2 x^2}{a^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b^2 x^2}{a^2}\right )}{\sqrt{-x} \sqrt{a-b x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 93, normalized size = 1.2 \begin{align*} -{\frac{a}{b \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{-bx+a}\sqrt{bx+a}\sqrt{{\frac{bx+a}{a}}}\sqrt{-2\,{\frac{bx-a}{a}}}\sqrt{-{\frac{bx}{a}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+a}{a}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{-x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a} \sqrt{-b x + a} \sqrt{-x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x)^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*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 + a} \sqrt{-b x + a} \sqrt{-x}}{b^{2} x^{3} - a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x)^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 12.1434, size = 95, normalized size = 1.23 \begin{align*} \frac{{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{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{b}} - \frac{{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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a} \sqrt{-b x + a} \sqrt{-x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x)^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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