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