3.2620 \(\int \frac{1}{\sqrt{a+b x} \sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

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

[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 = 0.205639, 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 \[ \frac{2 \sqrt{a} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b \sqrt{1-c}} \]

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])

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Rubi in Sympy [A]  time = 20.5572, size = 42, normalized size = 0.72 \[ \frac{2 \sqrt{a} F\left (\operatorname{asin}{\left (\frac{\sqrt{a + b x} \sqrt{- e + 1}}{\sqrt{a}} \right )}\middle | \frac{c - 1}{e - 1}\right )}{b \sqrt{- e + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

2*sqrt(a)*elliptic_f(asin(sqrt(a + b*x)*sqrt(-e + 1)/sqrt(a)), (c - 1)/(e - 1))/
(b*sqrt(-e + 1))

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Mathematica [B]  time = 0.469788, size = 129, normalized size = 2.22 \[ -\frac{2 (a+b x) \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{c-1}}}{\sqrt{a+b x}}\right )|\frac{c-1}{e-1}\right )}{b \sqrt{-\frac{a}{c-1}} \sqrt{\frac{b (c-1) x}{a}+c} \sqrt{\frac{b (e-1) x}{a}+e}} \]

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[Sqrt[-(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 = 0.253, size = 181, normalized size = 3.1 \[ -2\,{\frac{a \left ( c-e \right ) }{\sqrt{bx+a}b \left ( c-1 \right ) \left ( -1+e \right ) }\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ){\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}{\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}} \]

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. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{\frac{b{\left (c - 1\right )} x}{a} + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)),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. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a} \sqrt{\frac{a c +{\left (b c - b\right )} x}{a}} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="fricas")

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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