Integrand size = 81, antiderivative size = 526 \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\left (b-\sqrt {b^2-4 a c}\right ) x \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}+\frac {\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \]
x*(b-(-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)-1/2*(1/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2))) )^(1/2)*EllipticE(x*2^(1/2)*c^(1/2)/(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),(-2*(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/ 2)))^(1/2))*(b-(-4*a*c+b^2)^(1/2))*(1+2*c*x^2/(b-(-4*a*c+b^2)^(1/2)))^(1/2 )*(b+(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)/c^(1/2)/((1+2*c*x^2/(b-(-4*a*c+b^2) ^(1/2)))/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2))))^(1/2)+1/2*(1/(1+2*c*x^2/(b+(- 4*a*c+b^2)^(1/2))))^(1/2)*EllipticF(x*2^(1/2)*c^(1/2)/(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),(-2*(-4*a*c+b^2)^(1/2)/( b-(-4*a*c+b^2)^(1/2)))^(1/2))*(b-(-4*a*c+b^2)^(1/2))*(1+2*c*x^2/(b-(-4*a*c +b^2)^(1/2)))^(1/2)*(b+(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)/c^(1/2)/((1+2*c*x ^2/(b-(-4*a*c+b^2)^(1/2)))/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2))))^(1/2)
Result contains complex when optimal does not.
Time = 2.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.19 \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=-2 i \sqrt {2} a \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \]
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]
(-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])]
Time = 0.74 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {281, 324, 320, 388, 313}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \, dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \left (b-\sqrt {b^2-4 a c}\right ) \int \frac {\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{\sqrt {\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1}}dx\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \left (b-\sqrt {b^2-4 a c}\right ) \left (\int \frac {1}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1}}dx+\frac {2 c \int \frac {x^2}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1}}dx}{b-\sqrt {b^2-4 a c}}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \left (b-\sqrt {b^2-4 a c}\right ) \left (\frac {2 c \int \frac {x^2}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1}}dx}{b-\sqrt {b^2-4 a c}}+\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \left (b-\sqrt {b^2-4 a c}\right ) \left (\frac {2 c \left (\frac {x \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{2 c \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) \int \frac {\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{\left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^{3/2}}dx}{2 c}\right )}{b-\sqrt {b^2-4 a c}}+\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \left (b-\sqrt {b^2-4 a c}\right ) \left (\frac {2 c \left (\frac {x \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{2 c \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\right )}{b-\sqrt {b^2-4 a c}}+\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\right )\) |
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]
(b - Sqrt[b^2 - 4*a*c])*((2*c*(((b - Sqrt[b^2 - 4*a*c])*x*Sqrt[1 + (2*c*x^ 2)/(b - Sqrt[b^2 - 4*a*c])])/(2*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])]*EllipticE[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])])/(2 *Sqrt[2]*c^(3/2)*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 + S qrt[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])]))
3.1.56.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(2476\) vs. \(2(499)=998\).
Time = 1.37 (sec) , antiderivative size = 2477, normalized size of antiderivative = 4.71
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,method=_RETURNVERBOSE)
-1/2*(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(2* c*x^2+(-4*a*c+b^2)^(1/2)+b)/a/c)^(1/2)*((2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-2 *c*x^2+(-4*a*c+b^2)^(1/2)-b)*(4*a*c-b^2)/a/c)^(1/2)/((-2*c*x^2+(-4*a*c+b^2 )^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))^(1/2)/((2*c*x^2+(-4*a*c+b^2)^(1/2)+b)/ (b+(-4*a*c+b^2)^(1/2)))^(1/2)/(2*((2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-2*c*x^2 +(-4*a*c+b^2)^(1/2)-b)*(4*a*c-b^2)/a/c)^(1/2)*c*x^2+4*(-(-2*c*x^2+(-4*a*c+ b^2)^(1/2)-b)*(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)/a/c)^(1/2)*a*c-(-(-2*c*x^2+(- 4*a*c+b^2)^(1/2)-b)*(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)/a/c)^(1/2)*b^2+((2*c*x^ 2+(-4*a*c+b^2)^(1/2)+b)*(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(4*a*c-b^2)/a/c)^( 1/2)*b)*(1/2*b/(-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+ (-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4+2*((-4*a*c+b^2)^(3 /2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2 )^(1/2))/a*x^2)^(1/2)*(4-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a* b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(1+2*c*x^ 2/(b+(-4*a*c+b^2)^(1/2))+2*c*x^2/(b-(-4*a*c+b^2)^(1/2))+4*c^2/(b-(-4*a*c+b ^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))*x^4)^(1/2)*EllipticF(1/2*x*(-2*((-4*a*c+ b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4* a*c+b^2)^(1/2))/a)^(1/2),1/4*(-16-2*(2*c/(b+(-4*a*c+b^2)^(1/2))+2*c/(b-(-4 *a*c+b^2)^(1/2)))*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(-b+ (-4*a*c+b^2)^(1/2))/a/c^2*(b-(-4*a*c+b^2)^(1/2)))^(1/2))-2*c/(-2*((-4*a...
Time = 0.10 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.71 \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left (a c x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - a b x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} \sqrt {\frac {c}{a}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (\sqrt {b^{2} - 4 \, a c} b x - {\left (2 \, a b + b^{2}\right )} x + {\left ({\left (2 \, a - b\right )} c x + \sqrt {b^{2} - 4 \, a c} c x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} \sqrt {\frac {c}{a}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, a c \sqrt {\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}} \sqrt {\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}}}{2 \, c x} \]
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")
1/2*(2*sqrt(1/2)*(a*c*x*sqrt((b^2 - 4*a*c)/c^2) - a*b*x)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*sqrt(c/a)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt( (b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2* a*c)/(a*c)) - sqrt(1/2)*(sqrt(b^2 - 4*a*c)*b*x - (2*a*b + b^2)*x + ((2*a - b)*c*x + sqrt(b^2 - 4*a*c)*c*x)*sqrt((b^2 - 4*a*c)/c^2))*sqrt((c*sqrt((b^ 2 - 4*a*c)/c^2) - b)/c)*sqrt(c/a)*elliptic_f(arcsin(sqrt(1/2)*sqrt((c*sqrt ((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2 *a*c)/(a*c)) + 2*a*c*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)
\[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int \frac {b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{\sqrt {\frac {b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{b - \sqrt {- 4 a c + b^{2}}}} \sqrt {\frac {b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{b + \sqrt {- 4 a c + b^{2}}}}}\, dx \]
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)
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)
\[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int { \frac {2 \, c x^{2} + b - \sqrt {b^{2} - 4 \, a c}}{\sqrt {\frac {2 \, c x^{2}}{b + \sqrt {b^{2} - 4 \, a c}} + 1} \sqrt {\frac {2 \, c x^{2}}{b - \sqrt {b^{2} - 4 \, a c}} + 1}} \,d x } \]
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")
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)
\[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int { \frac {2 \, c x^{2} + b - \sqrt {b^{2} - 4 \, a c}}{\sqrt {\frac {2 \, c x^{2}}{b + \sqrt {b^{2} - 4 \, a c}} + 1} \sqrt {\frac {2 \, c x^{2}}{b - \sqrt {b^{2} - 4 \, a c}} + 1}} \,d x } \]
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")
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)
Timed out. \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int \frac {b+2\,c\,x^2-\sqrt {b^2-4\,a\,c}}{\sqrt {\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}+1}\,\sqrt {\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}+1}} \,d x \]
int((b + 2*c*x^2 - (b^2 - 4*a*c)^(1/2))/(((2*c*x^2)/(b - (b^2 - 4*a*c)^(1/ 2)) + 1)^(1/2)*((2*c*x^2)/(b + (b^2 - 4*a*c)^(1/2)) + 1)^(1/2)),x)