\(\int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx\) [299]

Optimal result

Integrand size = 59, antiderivative size = 478 \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {x \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}-\frac {\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \]

[Out]

x*(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))*(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))*(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)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {433, 429, 506, 422} \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}-\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}+\frac {x \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{\sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]

[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]

[Out]

(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])] - (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])])/(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])]) + (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 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[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]

Rule 429

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

Rule 433

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 506

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] - 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]

Rubi steps \begin{align*} \text {integral}& = \frac {(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}{b-\sqrt {b^2-4 a c}}+\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 {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 {\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}}}}-\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 {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 {\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 {\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 [A] (verified)

Time = 2.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\sqrt {-b-\sqrt {b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \]

[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]

[Out]

(Sqrt[-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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1387\) vs. \(2(457)=914\).

Time = 2.69 (sec) , antiderivative size = 1388, normalized size of antiderivative = 2.90

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1388\)

[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,method=_RETURNVERBOSE)

[Out]

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)*(-b+(-4*a*c+b^2)^(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)/a
/c)^(1/2)/(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(1/2/(-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/(-b+(-4*a*c+b^2)^(1/2))/(-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)/(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)^(1/2)/a)*(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))-EllipticE(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))))

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (b^{2} x + \sqrt {b^{2} - 4 \, a c} b x - {\left (b c x + \sqrt {b^{2} - 4 \, a c} c x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} \sqrt {\frac {c}{a}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (\sqrt {b^{2} - 4 \, a c} b x + {\left (b^{2} + 2 \, b c\right )} x - {\left (\sqrt {b^{2} - 4 \, a c} c x + {\left (b c - 2 \, c^{2}\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} \sqrt {\frac {c}{a}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} \sqrt {\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}} \sqrt {\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}}}{4 \, c^{2} x} \]

[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, algorithm="fric
as")

[Out]

-1/4*(sqrt(1/2)*(b^2*x + sqrt(b^2 - 4*a*c)*b*x - (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_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 + (b^2 + 2*b
*c)*x - (sqrt(b^2 - 4*a*c)*c*x + (b*c - 2*c^2)*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)) - (b*c + sqrt(b^2 - 4*a*c)*c)*sqrt((b*x^2 + sqrt(b^2 - 4*a*c)*x^2 + 2*a)/a)*sq
rt((b*x^2 - sqrt(b^2 - 4*a*c)*x^2 + 2*a)/a))/(c^2*x)

Sympy [F]

\[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int \frac {\sqrt {\frac {b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{b - \sqrt {- 4 a c + b^{2}}}}}{\sqrt {\frac {b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{b + \sqrt {- 4 a c + b^{2}}}}}\, dx \]

[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)

[Out]

Integral(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)

Maxima [F]

\[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int { \frac {\sqrt {\frac {2 \, c x^{2}}{b - \sqrt {b^{2} - 4 \, a c}} + 1}}{\sqrt {\frac {2 \, c x^{2}}{b + \sqrt {b^{2} - 4 \, a c}} + 1}} \,d x } \]

[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, algorithm="maxi
ma")

[Out]

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)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\text {Exception raised: TypeError} \]

[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, algorithm="giac
")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\int \frac {\sqrt {\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}+1}}{\sqrt {\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}+1}} \,d x \]

[In]

int(((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)

[Out]

int(((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)