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

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

[Out]

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

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Rubi [A]  time = 0.0640603, 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 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )\right |-1\right )}{\sqrt{b} \sqrt{c}} \]

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])

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Rubi in Sympy [A]  time = 5.95117, size = 32, normalized size = 0.97 \[ \frac{2 F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}} \right )}\middle | -1\right )}{\sqrt{b} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.131154, size = 76, normalized size = 2.3 \[ -\frac{2 i \sqrt{-\frac{1}{c}} c x^{3/2} \sqrt{1-\frac{1}{c^2 x^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{b x} \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[-c^(-1)]*c*Sqrt[1 - 1/(c^2*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-
c^(-1)]/Sqrt[x]], -1])/(Sqrt[b*x]*Sqrt[1 - c^2*x^2])

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Maple [A]  time = 0.085, size = 32, normalized size = 1. \[{\frac{\sqrt{2}}{c}\sqrt{-cx}{\it EllipticF} \left ( \sqrt{cx+1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{bx}}}} \]

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. \[ \int \frac{1}{\sqrt{b x} \sqrt{c x + 1} \sqrt{-c x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)),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. \[{\rm integral}\left (\frac{1}{\sqrt{b x} \sqrt{c x + 1} \sqrt{-c x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 39.3265, size = 94, normalized size = 2.85 \[ \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{b} \sqrt{c}} - \frac{i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{b} \sqrt{c}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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