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 = 0.203864, 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, Rules used = {21, 424} \[ -\frac{\sqrt{b-\sqrt{b^2-4 a c}} \left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]

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]))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \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 &=\left (-b-\sqrt{b^2-4 a c}\right ) \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\\ &=-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \left (b+\sqrt{b^2-4 a c}\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}}\\ \end{align*}

Mathematica [C]  time = 0.383008, 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]*Sqrt[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 = 0.236, size = 0, normalized size = 0. \begin{align*} \int{ \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 \end{align*}

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. \begin{align*} \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} \end{align*}

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, 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 - sqr
t(b^2 - 4*a*c)) + 1)), x)

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

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, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="giac")

[Out]

Timed out