3.2623 \(\int \frac{\sqrt{e+\frac{b (-1+e) x}{a}}}{\sqrt{a+b x} \sqrt{c+\frac{b (-1+c) x}{a}}} \, dx\)

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

[Out]

(2*Sqrt[a]*EllipticE[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 = 0.17513, 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 \[ \frac{2 \sqrt{a} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b \sqrt{1-c}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*EllipticE[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 = 18.2144, size = 42, normalized size = 0.72 \[ \frac{2 \sqrt{a} E\left (\operatorname{asin}{\left (\frac{\sqrt{a + b x} \sqrt{- c + 1}}{\sqrt{a}} \right )}\middle | \frac{e - 1}{c - 1}\right )}{b \sqrt{- c + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.14863, size = 191, normalized size = 3.29 \[ -\frac{2 (a+b x)^{3/2} \left (\frac{a \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{e-1}{c-1}\right )}{\sqrt{a+b x}}-\frac{\sqrt{-\frac{a}{e-1}} \left (\frac{a}{a+b x}+c-1\right ) \left (\frac{a}{a+b x}+e-1\right )}{c-1}\right )}{a b \sqrt{-\frac{a}{e-1}} \sqrt{\frac{b (c-1) x}{a}+c} \sqrt{\frac{b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x)^(3/2)*(-((Sqrt[-(a/(-1 + e))]*(-1 + c + a/(a + b*x))*(-1 + e + a/(
a + b*x)))/(-1 + c)) + (a*Sqrt[(-1 + c + a/(a + b*x))/(-1 + c)]*Sqrt[(-1 + e + a
/(a + b*x))/(-1 + e)]*EllipticE[ArcSin[Sqrt[-(a/(-1 + e))]/Sqrt[a + b*x]], (-1 +
e)/(-1 + c)])/Sqrt[a + b*x]))/(a*b*Sqrt[-(a/(-1 + e))]*Sqrt[c + (b*(-1 + c)*x)/
a]*Sqrt[e + (b*(-1 + e)*x)/a])

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Maple [B]  time = 0.053, size = 548, normalized size = 9.5 \[ -2\,{\frac{{a}^{2}\sqrt{bx+a}}{ \left ({b}^{2}e{x}^{2}+2\,abex-{b}^{2}{x}^{2}+{a}^{2}e-abx \right ) \left ( c-1 \right ) ^{2}b \left ( -1+e \right ) }\sqrt{{\frac{bxe+ae-bx}{a}}}\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}} \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ){c}^{2}-{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ce-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ce+{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ){e}^{2}-{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) c+{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) e+{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) c-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) e \right ){\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}{\sqrt{b x + a} \sqrt{\frac{a c +{\left (b c - b\right )} x}{a}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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