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

Optimal. Leaf size=37 \[ -\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{-b}}\right )\right |-1\right )}{\sqrt{-b} \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0812523, antiderivative size = 37, 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 \[ -\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{-b}}\right )\right |-1\right )}{\sqrt{-b} \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.79607, size = 37, normalized size = 1. \[ - \frac{2 E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{- b}} \right )}\middle | -1\right )}{\sqrt{c} \sqrt{- b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.846207, size = 77, normalized size = 2.08 \[ \frac{2 c \left (-\sqrt{\frac{1}{c}} x^{3/2} \sqrt{1-\frac{1}{c^2 x^2}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{1}{c}}}{\sqrt{x}}\right )\right |-1\right )+\frac{1}{c^2}-x^2\right )}{\sqrt{b x} \sqrt{1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*c*(c^(-2) - x^2 - Sqrt[c^(-1)]*Sqrt[1 - 1/(c^2*x^2)]*x^(3/2)*EllipticE[ArcSin
[Sqrt[c^(-1)]/Sqrt[x]], -1]))/(Sqrt[b*x]*Sqrt[1 - c^2*x^2])

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Maple [A]  time = 0.02, size = 33, normalized size = 0.9 \[ 2\,{\frac{\sqrt{2}\sqrt{-cx}{\it EllipticE} \left ( \sqrt{cx+1},1/2\,\sqrt{2} \right ) }{c\sqrt{bx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*2^(1/2)*(-c*x)^(1/2)*EllipticE((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. \[ \int \frac{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{c x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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