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

Optimal. Leaf size=38 \[ \frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{b}}\right )|-\frac{c}{d}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0096796, antiderivative size = 38, 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 = 0.354174, size = 102, normalized size = 2.68 \[ \frac{2 \sqrt{1-d x} \left (\frac{\sqrt{x} \sqrt{\frac{1}{c x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right )|-\frac{c}{d}\right )}{\sqrt{-\frac{1}{c}} \sqrt{1-\frac{1}{d x}}}-c x-1\right )}{d \sqrt{b x} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.042, size = 129, normalized size = 3.4 \begin{align*} -2\,{\frac{\sqrt{-cx}\sqrt{-dx+1}}{ \left ( dx-1 \right ) \sqrt{bx}cd} \left ({\it EllipticF} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) c+{\it EllipticF} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) d-{\it EllipticE} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) c-{\it EllipticE} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) d \right ) \sqrt{-{\frac{ \left ( dx-1 \right ) c}{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*(EllipticF((c*x+1)^(1/2),(d/(c+d))^(1/2))*c+EllipticF((c*x+1)^(1/2),(d/(c+d))^(1/2))*d-EllipticE((c*x+1)^(1
/2),(d/(c+d))^(1/2))*c-EllipticE((c*x+1)^(1/2),(d/(c+d))^(1/2))*d)*(-c*x)^(1/2)*(-(d*x-1)*c/(c+d))^(1/2)*(-d*x
+1)^(1/2)/(d*x-1)/(b*x)^(1/2)/c/d

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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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

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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)