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

Optimal. Leaf size=95 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]

[Out]

(Sqrt[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]*Sq
rt[c])

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Rubi [A]  time = 0.297252, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.017 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]

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]

(Sqrt[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]*Sq
rt[c])

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Rubi in Sympy [A]  time = 23.6938, size = 87, normalized size = 0.92 \[ \frac{\sqrt{2} \sqrt{b + \sqrt{- 4 a c + b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}\middle | - \frac{b + \sqrt{- 4 a c + b^{2}}}{b - \sqrt{- 4 a c + b^{2}}}\right )}{2 \sqrt{c}} \]

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)*sqrt(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))

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Mathematica [A]  time = 0.413655, size = 92, normalized size = 0.97 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}} \]

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 = 0.319, 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)))/sq
rt(-(-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