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

Optimal. Leaf size=113 \[ -\frac{\sqrt{b-\sqrt{b^2-4 a c}} \left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]

[Out]

-((Sqrt[b - Sqrt[b^2 - 4*a*c]]*(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]))

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Rubi [A]  time = 0.488136, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 87, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023 \[ -\frac{\sqrt{b-\sqrt{b^2-4 a c}} \left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 47.7069, size = 100, normalized size = 0.88 \[ - \frac{\sqrt{2} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (b + \sqrt{- 4 a c + b^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}\middle | \frac{b - \sqrt{- 4 a c + b^{2}}}{b + \sqrt{- 4 a c + b^{2}}}\right )}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b+2*c*x**2-(-4*a*c+b**2)**(1/2))/(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))*(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 [C]  time = 0.661414, size = 104, normalized size = 0.92 \[ -2 i \sqrt{2} a \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right ) \]

Antiderivative was successfully verified.

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

(-2*I)*Sqrt[2]*a*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sq
rt[c/(-b + Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c
])]

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Maple [F]  time = 0.29, size = 0, normalized size = 0. \[ \int{1 \left ( 2\,c{x}^{2}-\sqrt{-4\,ac+{b}^{2}}-b \right ){\frac{1}{\sqrt{1+2\,{\frac{c{x}^{2}}{-b-\sqrt{-4\,ac+{b}^{2}}}}}}}{\frac{1}{\sqrt{1+2\,{\frac{c{x}^{2}}{-b+\sqrt{-4\,ac+{b}^{2}}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x^2-(-4*a*c+b^2)^(1/2)-b)/(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((2*c*x^2-(-4*a*c+b^2)^(1/2)-b)/(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. \[ \int \frac{2 \, c x^{2} - b - \sqrt{b^{2} - 4 \, a c}}{\sqrt{-\frac{2 \, c x^{2}}{b + \sqrt{b^{2} - 4 \, a c}} + 1} \sqrt{-\frac{2 \, c x^{2}}{b - \sqrt{b^{2} - 4 \, a c}} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 - b - sqrt(b^2 - 4*a*c))/(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((2*c*x^2 - b - sqrt(b^2 - 4*a*c))/(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. \[{\rm integral}\left (\frac{2 \, c x^{2} - b - \sqrt{b^{2} - 4 \, a c}}{\sqrt{-\frac{2 \, c x^{2} - b - \sqrt{b^{2} - 4 \, a c}}{b + \sqrt{b^{2} - 4 \, a c}}} \sqrt{-\frac{2 \, c x^{2} - b + \sqrt{b^{2} - 4 \, a c}}{b - \sqrt{b^{2} - 4 \, a c}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 - b - sqrt(b^2 - 4*a*c))/(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((2*c*x^2 - 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)))*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. \[ \int \frac{- b + 2 c x^{2} - \sqrt{- 4 a c + b^{2}}}{\sqrt{\frac{- b + 2 c x^{2} - \sqrt{- 4 a c + b^{2}}}{- b - \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{- b + 2 c x^{2} + \sqrt{- 4 a c + b^{2}}}{- b + \sqrt{- 4 a c + b^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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