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