3.2637 \(\int \frac{1}{\sqrt{a+b x} \sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=58 \[ \frac{2 \sqrt{a} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right ),\frac{1-e}{1-c}\right )}{b \sqrt{1-c}} \]

[Out]

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

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Rubi [A]  time = 0.0366825, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.025, Rules used = {119} \[ \frac{2 \sqrt{a} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b \sqrt{1-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.303974, size = 129, normalized size = 2.22 \[ -\frac{2 (a+b x) \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{c-1}}}{\sqrt{a+b x}}\right ),\frac{c-1}{e-1}\right )}{b \sqrt{-\frac{a}{c-1}} \sqrt{\frac{b (c-1) x}{a}+c} \sqrt{\frac{b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.144, size = 181, normalized size = 3.1 \begin{align*} -2\,{\frac{a \left ( c-e \right ) }{\sqrt{bx+a}b \left ( c-1 \right ) \left ( -1+e \right ) }\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ){\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}{\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)/(c+b*(c-1)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x)

[Out]

-2/(b*x+a)^(1/2)*a*(-(-1+e)*(b*c*x+a*c-b*x)/a/(c-e))^(1/2)*(-(b*x+a)*(c-1)/a)^(1/2)*((c-1)*(b*e*x+a*e-b*x)/a/(
c-e))^(1/2)*EllipticF((-(-1+e)*(b*c*x+a*c-b*x)/a/(c-e))^(1/2),(-(c-e)/(-1+e))^(1/2))*(c-e)/((b*c*x+a*c-b*x)/a)
^(1/2)/((b*e*x+a*e-b*x)/a)^(1/2)/b/(c-1)/(-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{\frac{b{\left (c - 1\right )} x}{a} + 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)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + 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} a^{2} \sqrt{\frac{a c +{\left (b c - b\right )} x}{a}} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}{a^{3} c e -{\left (b^{3} c - b^{3} -{\left (b^{3} c - b^{3}\right )} e\right )} x^{3} -{\left (2 \, a b^{2} c - a b^{2} -{\left (3 \, a b^{2} c - 2 \, a b^{2}\right )} e\right )} x^{2} -{\left (a^{2} b c -{\left (3 \, a^{2} b c - a^{2} b\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)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*a^2*sqrt((a*c + (b*c - b)*x)/a)*sqrt((a*e + (b*e - b)*x)/a)/(a^3*c*e - (b^3*c - b^3 - (
b^3*c - b^3)*e)*x^3 - (2*a*b^2*c - a*b^2 - (3*a*b^2*c - 2*a*b^2)*e)*x^2 - (a^2*b*c - (3*a^2*b*c - a^2*b)*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 + \frac{b c x}{a} - \frac{b x}{a}} \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)/(c+b*(-1+c)*x/a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x)*sqrt(c + b*c*x/a - b*x/a)*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{\frac{b{\left (c - 1\right )} x}{a} + 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)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="giac")

[Out]

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