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

Optimal. Leaf size=526 \[ \frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ),-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{x \left (b-\sqrt{b^2-4 a c}\right ) \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}}{\sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

((b - Sqrt[b^2 - 4*a*c])*x*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])] - ((b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Ellip
ticE[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])
/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(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])]) + ((b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^
2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 -
4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(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])])

________________________________________________________________________________________

Rubi [A]  time = 0.630035, antiderivative size = 526, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 81, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {21, 422, 418, 492, 411} \[ \frac{x \left (b-\sqrt{b^2-4 a c}\right ) \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}}{\sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

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 + Sqr
t[b^2 - 4*a*c])]),x]

[Out]

((b - Sqrt[b^2 - 4*a*c])*x*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])] - ((b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Ellip
ticE[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])
/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(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])]) + ((b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^
2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*Sqrt[b^2 -
4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))/(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])])

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 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[a, Int[1/(Sqrt[a + b*x^2]*Sqrt[c +
d*x^2]), x], x] + Dist[b, Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[
d/c] && PosQ[b/a]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

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\\ &=(2 c) \int \frac{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{1}{\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{\left (b-\sqrt{b^2-4 a c}\right ) x \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}}}}+\frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}{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}}}}+\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}}}}{\left (1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )^{3/2}} \, dx\\ &=\frac{\left (b-\sqrt{b^2-4 a c}\right ) x \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}}}}-\frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}{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}}}}+\frac{\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}{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}}}}\\ \end{align*}

Mathematica [C]  time = 0.4025, size = 203, normalized size = 0.39 \[ -\frac{i \left (\left (\sqrt{b^2-4 a c}+b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b-\sqrt{b^2-4 a c}}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )-2 \sqrt{b^2-4 a c} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{b-\sqrt{b^2-4 a c}}}\right ),\frac{b-\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}+b}\right )\right )}{\sqrt{2} \sqrt{\frac{c}{b-\sqrt{b^2-4 a c}}}} \]

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]

((-I)*((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])] - 2*Sqrt[b^2 - 4*a*c]*EllipticF[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])]))/(Sqrt[2]*Sqrt[c/(b - Sqrt[b^2 - 4*a*c])])

________________________________________________________________________________________

Maple [F]  time = 0.154, 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)

________________________________________________________________________________________

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((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, 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)

________________________________________________________________________________________

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((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, 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)

________________________________________________________________________________________

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((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]

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)

________________________________________________________________________________________

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((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, algorithm="giac")

[Out]

Timed out