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

Optimal. Leaf size=42 \[ -\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0168393, antiderivative size = 42, 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} \[ -\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.411434, size = 112, normalized size = 2.67 \[ \frac{-2 x^{3/2} \sqrt{1-\frac{1}{c x}} \sqrt{\frac{1}{d x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{c}}}{\sqrt{x}}\right )|-\frac{c}{d}\right )-\frac{2 \sqrt{\frac{1}{c}} (c x-1) (d x+1)}{d}}{\sqrt{\frac{1}{c}} \sqrt{b x} \sqrt{1-c x} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 67, normalized size = 1.6 \begin{align*} -2\,{\frac{ \left ( c+d \right ) \sqrt{-dx}\sqrt{-cx+1}}{ \left ( cx-1 \right ) \sqrt{bx}{d}^{2}}{\it EllipticE} \left ( \sqrt{dx+1},\sqrt{{\frac{c}{c+d}}} \right ) \sqrt{-{\frac{ \left ( cx-1 \right ) d}{c+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

Integral(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*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{d 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)/(d*x+1)^(1/2),x, algorithm="giac")

[Out]

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