3.2625 \(\int \frac{\sqrt{a+b x}}{\sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}} \]

[Out]

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

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Rubi [A]  time = 0.824337, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/(c+b*(-1+c)*x/a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.295552, size = 103, normalized size = 0.64 \[ -\frac{2 i a \sqrt{a+b x} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{(c-1) (a+b x)}{a}}\right )|\frac{e-1}{c-1}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{(c-1) (a+b x)}{a}}\right )|\frac{e-1}{c-1}\right )\right )}{b (e-1) \sqrt{\frac{(c-1) (a+b x)}{a}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.052, size = 183, normalized size = 1.1 \[ 2\,{\frac{ \left ( c-e \right ){a}^{2}}{\sqrt{bx+a} \left ( c-1 \right ) ^{2}b \left ( -1+e \right ) }{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) \sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}{\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}{\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(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. \[ \int \frac{\sqrt{a + b x}}{\sqrt{c + \frac{b c x}{a} - \frac{b x}{a}} \sqrt{e + \frac{b e x}{a} - \frac{b x}{a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/(c+b*(-1+c)*x/a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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