3.297 \(\int \frac{\sqrt{1-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx\)

Optimal. Leaf size=215 \[ \frac{\sqrt{2} b F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}} \]

[Out]

-(((b + Sqrt[b^2 - 4*a*c])*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^
2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[2]*Sqrt
[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*b*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]
*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*
c]))])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.661834, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{\sqrt{2} b F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]/Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])],x]

[Out]

-(((b + Sqrt[b^2 - 4*a*c])*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^
2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[2]*Sqrt
[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*b*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]
*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*
c]))])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 82.5352, size = 185, normalized size = 0.86 \[ \frac{\sqrt{2} b F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}\middle | \frac{- b + \sqrt{- 4 a c + b^{2}}}{b + \sqrt{- 4 a c + b^{2}}}\right )}{\sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}}} - \frac{\sqrt{2} \left (b + \sqrt{- 4 a c + b^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}\middle | \frac{- b + \sqrt{- 4 a c + b^{2}}}{b + \sqrt{- 4 a c + b^{2}}}\right )}{2 \sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*c*x**2/(b-(-4*a*c+b**2)**(1/2)))**(1/2)/(1+2*c*x**2/(b+(-4*a*c+b**2)**(1/2)))**(1/2),x)

[Out]

sqrt(2)*b*elliptic_f(asin(sqrt(2)*sqrt(c)*x/sqrt(b - sqrt(-4*a*c + b**2))), (-b
+ sqrt(-4*a*c + b**2))/(b + sqrt(-4*a*c + b**2)))/(sqrt(c)*sqrt(b - sqrt(-4*a*c
+ b**2))) - sqrt(2)*(b + sqrt(-4*a*c + b**2))*elliptic_e(asin(sqrt(2)*sqrt(c)*x/
sqrt(b - sqrt(-4*a*c + b**2))), (-b + sqrt(-4*a*c + b**2))/(b + sqrt(-4*a*c + b*
*2)))/(2*sqrt(c)*sqrt(b - sqrt(-4*a*c + b**2)))

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Mathematica [A]  time = 0.313601, size = 94, normalized size = 0.44 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{2} \sqrt{-\frac{c}{\sqrt{b^2-4 a c}+b}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]/Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])],x]

[Out]

EllipticE[ArcSin[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))]*x], (b + Sqrt[b^2 -
4*a*c])/(-b + Sqrt[b^2 - 4*a*c])]/(Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))])

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int{1\sqrt{1-2\,{\frac{c{x}^{2}}{b-\sqrt{-4\,ac+{b}^{2}}}}}{\frac{1}{\sqrt{1+2\,{\frac{c{x}^{2}}{b+\sqrt{-4\,ac+{b}^{2}}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*c*x^2/(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^(1/2),x)

[Out]

int((1-2*c*x^2/(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^
(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-\frac{2 \, c x^{2}}{b - \sqrt{b^{2} - 4 \, a c}} + 1}}{\sqrt{\frac{2 \, c x^{2}}{b + \sqrt{b^{2} - 4 \, a c}} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*c*x^2/(b - sqrt(b^2 - 4*a*c)) + 1)/sqrt(2*c*x^2/(b + sqrt(b^2 - 4*a*c)) + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(-2*c*x^2/(b - sqrt(b^2 - 4*a*c)) + 1)/sqrt(2*c*x^2/(b + sqrt(b^2
- 4*a*c)) + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-\frac{2 \, c x^{2} - b + \sqrt{b^{2} - 4 \, a c}}{b - \sqrt{b^{2} - 4 \, a c}}}}{\sqrt{\frac{2 \, c x^{2} + b + \sqrt{b^{2} - 4 \, a c}}{b + \sqrt{b^{2} - 4 \, a c}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*c*x^2/(b - sqrt(b^2 - 4*a*c)) + 1)/sqrt(2*c*x^2/(b + sqrt(b^2 - 4*a*c)) + 1),x, algorithm="fricas")

[Out]

integral(sqrt(-(2*c*x^2 - b + sqrt(b^2 - 4*a*c))/(b - sqrt(b^2 - 4*a*c)))/sqrt((
2*c*x^2 + b + sqrt(b^2 - 4*a*c))/(b + sqrt(b^2 - 4*a*c))), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \frac{- b + 2 c x^{2} + \sqrt{- 4 a c + b^{2}}}{b - \sqrt{- 4 a c + b^{2}}}}}{\sqrt{\frac{b + 2 c x^{2} + \sqrt{- 4 a c + b^{2}}}{b + \sqrt{- 4 a c + b^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*c*x**2/(b-(-4*a*c+b**2)**(1/2)))**(1/2)/(1+2*c*x**2/(b+(-4*a*c+b**2)**(1/2)))**(1/2),x)

[Out]

Integral(sqrt(-(-b + 2*c*x**2 + sqrt(-4*a*c + b**2))/(b - sqrt(-4*a*c + b**2)))/
sqrt((b + 2*c*x**2 + sqrt(-4*a*c + b**2))/(b + sqrt(-4*a*c + b**2))), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*c*x^2/(b - sqrt(b^2 - 4*a*c)) + 1)/sqrt(2*c*x^2/(b + sqrt(b^2 - 4*a*c)) + 1),x, algorithm="giac")

[Out]

Timed out