3.863 \(\int \frac{1}{\sqrt{b x} \sqrt{1-c x} \sqrt{1+c x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0109767, 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} \[ \frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )\right |-1\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.011276, size = 27, normalized size = 0.82 \[ \frac{2 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right )}{\sqrt{b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 32, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{c}\sqrt{-cx}{\it EllipticF} \left ( \sqrt{cx+1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{bx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b x} \sqrt{c x + 1} \sqrt{-c x + 1}}{b c^{2} x^{3} - b x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(b*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(b*c^2*x^3 - b*x), x)

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Sympy [B]  time = 12.9649, size = 94, normalized size = 2.85 \begin{align*} \frac{i{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{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{b} \sqrt{c}} - \frac{i{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{e^{- 2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{b} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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