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

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

[Out]

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

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Rubi [A]  time = 0.0089643, 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 = {116} \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )|-\frac{d}{c}\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

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

Mathematica [B]  time = 0.124206, size = 89, normalized size = 2.34 \[ -\frac{2 x^{3/2} \sqrt{\frac{c-\frac{1}{x}}{c}} \sqrt{\frac{d+\frac{1}{x}}{d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{c}}}{\sqrt{x}}\right ),-\frac{c}{d}\right )}{\sqrt{\frac{1}{c}} \sqrt{b x} \sqrt{1-c x} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(c - x^(-1))/c]*Sqrt[(d + x^(-1))/d]*x^(3/2)*EllipticF[ArcSin[Sqrt[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 [B]  time = 0.058, size = 64, normalized size = 1.7 \begin{align*} -2\,{\frac{\sqrt{-cx+1}\sqrt{-dx}}{\sqrt{bx} \left ( cx-1 \right ) d}\sqrt{-{\frac{ \left ( cx-1 \right ) d}{c+d}}}{\it EllipticF} \left ( \sqrt{dx+1},\sqrt{{\frac{c}{c+d}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x)*sqrt(-c*x + 1)*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 c d x^{3} +{\left (b c - b d\right )} x^{2} - b x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(b*x)*sqrt(-c*x + 1)*sqrt(d*x + 1)/(b*c*d*x^3 + (b*c - b*d)*x^2 - b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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