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

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 = 0.0217859, 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, Rules used = {110} \[ -\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{-b}}\right )\right |-1\right )}{\sqrt{-b} \sqrt{c}} \]

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])

Rule 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, x] /; FreeQ[{b, c, d, e, f}, x] &&
NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-(b/d), 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-c x}}{\sqrt{b x} \sqrt{1+c x}} \, dx &=-\frac{2 E\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 = 0.0316616, size = 52, normalized size = 1.41 \[ -\frac{2 x \left (c x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right )\right )}{3 \sqrt{b x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*x*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2] + c*x*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2]))/(3*Sqrt[
b*x])

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Maple [A]  time = 0.012, size = 33, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{2}\sqrt{-cx}{\it EllipticE} \left ( \sqrt{cx+1},1/2\,\sqrt{2} \right ) }{c\sqrt{bx}}} \end{align*}

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

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x} \sqrt{c x + 1} \sqrt{-c x + 1}}{b c x^{2} + b x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

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

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

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]

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

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

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, algorithm="giac")

[Out]

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