3.2638 \(\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right ),-\frac{a d}{(1-e) (b c-a d)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]

[Out]

(2*Sqrt[a]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(Sqrt[1 - e]*Sqrt[a + b*x])/Sqrt[a]], -((a*d)/((b*
c - a*d)*(1 - e)))])/(b*Sqrt[1 - e]*Sqrt[c + d*x])

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Rubi [A]  time = 0.0745662, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {121, 119} \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(Sqrt[1 - e]*Sqrt[a + b*x])/Sqrt[a]], -((a*d)/((b*
c - a*d)*(1 - e)))])/(b*Sqrt[1 - e]*Sqrt[c + d*x])

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 119

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

Rubi steps

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

Mathematica [A]  time = 0.405816, size = 126, normalized size = 1.31 \[ -\frac{2 \sqrt{c+d x} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{e-1}}}{\sqrt{a+b x}}\right ),\frac{(e-1) (b c-a d)}{a d}\right )}{d \sqrt{-\frac{a}{e-1}} \sqrt{\frac{b (e-1) x}{a}+e} \sqrt{\frac{b (c+d x)}{d (a+b x)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c + d*x]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticF[ArcSin[Sqrt[-(a/(-1 + e))]/Sqrt[a + b*x]], (
(b*c - a*d)*(-1 + e))/(a*d)])/(d*Sqrt[-(a/(-1 + e))]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + (b*(-1 + e)*x)
/a])

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Maple [B]  time = 0.107, size = 207, normalized size = 2.2 \begin{align*} 2\,{\frac{\sqrt{bx+a}\sqrt{dx+c} \left ( ade-bce+bc \right ) }{ \left ( d{x}^{2}b+adx+bcx+ac \right ) bd \left ( -1+e \right ) }\sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+e \right ) }{a}}}\sqrt{-{\frac{ \left ( dx+c \right ) b \left ( -1+e \right ) }{ade-bce+bc}}}{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} \sqrt{d x + c} a \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}{a^{2} c e +{\left (b^{2} d e - b^{2} d\right )} x^{3} -{\left (b^{2} c + a b d -{\left (b^{2} c + 2 \, a b d\right )} e\right )} x^{2} -{\left (a b c -{\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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