\(\int \frac {1}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx\) [295]

Optimal result

Integrand size = 59, antiderivative size = 98 \[ \int \frac {1}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx=\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b+\sqrt {b^2+4 a c}}}\right ),\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c}} \] Output:

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

Mathematica [A] (verified)

Time = 2.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx=\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b+\sqrt {b^2+4 a c}}}\right ),\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c}} \] Input:

Integrate[1/(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]
 

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(98)=196\).

Time = 0.33 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.12, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {320}

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.

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {\sqrt {4 a c+b^2}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}\)

Input:

Int[1/(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]
 

Output:

(Sqrt[b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*E 
llipticF[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])])
 

Defintions of rubi rules used

rule 320

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(80)=160\).

Time = 1.16 (sec) , antiderivative size = 610, normalized size of antiderivative = 6.22

method	result	size
elliptic	\(\frac {\sqrt {\frac {\left (-2 c \,x^{2}+\sqrt {4 a c +b^{2}}-b \right ) \left (2 c \,x^{2}+\sqrt {4 a c +b^{2}}+b \right )}{a c}}\, \sqrt {2}\, \sqrt {4-\frac {2 \left (\left (4 a c +b^{2}\right )^{\frac {3}{2}}-\sqrt {4 a c +b^{2}}\, b^{2}+4 a b c \right ) x^{2}}{\left (-b +\sqrt {4 a c +b^{2}}\right ) \left (b +\sqrt {4 a c +b^{2}}\right ) a}}\, \sqrt {4+\frac {2 \left (\left (4 a c +b^{2}\right )^{\frac {3}{2}}-\sqrt {4 a c +b^{2}}\, b^{2}-4 a b c \right ) x^{2}}{\left (-b +\sqrt {4 a c +b^{2}}\right ) \left (b +\sqrt {4 a c +b^{2}}\right ) a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {\left (4 a c +b^{2}\right )^{\frac {3}{2}}-\sqrt {4 a c +b^{2}}\, b^{2}+4 a b c}{\left (-b +\sqrt {4 a c +b^{2}}\right ) \left (b +\sqrt {4 a c +b^{2}}\right ) a}}}{2}, \frac {\sqrt {-16+\frac {2 \left (\frac {2 c}{b +\sqrt {4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {4 a c +b^{2}}}\right ) \left (\left (4 a c +b^{2}\right )^{\frac {3}{2}}-\sqrt {4 a c +b^{2}}\, b^{2}-4 a b c \right ) \left (b -\sqrt {4 a c +b^{2}}\right )}{\left (-b +\sqrt {4 a c +b^{2}}\right ) a \,c^{2}}}}{4}\right )}{8 \sqrt {\frac {-2 c \,x^{2}+\sqrt {4 a c +b^{2}}-b}{-b +\sqrt {4 a c +b^{2}}}}\, \sqrt {\frac {2 c \,x^{2}+\sqrt {4 a c +b^{2}}+b}{b +\sqrt {4 a c +b^{2}}}}\, \sqrt {\frac {\left (4 a c +b^{2}\right )^{\frac {3}{2}}-\sqrt {4 a c +b^{2}}\, b^{2}+4 a b c}{\left (-b +\sqrt {4 a c +b^{2}}\right ) \left (b +\sqrt {4 a c +b^{2}}\right ) a}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {4 a c +b^{2}}}+\frac {2 c \,x^{2}}{b -\sqrt {4 a c +b^{2}}}+\frac {4 c^{2} x^{4}}{\left (b -\sqrt {4 a c +b^{2}}\right ) \left (b +\sqrt {4 a c +b^{2}}\right )}}}\)	\(610\)

Input:

int(1/(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)
 

Output:

1/8/((-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*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^(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))/(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*x^4/(b-(4*a*c+b^2)^(1/2))/(b+(4*a*c+b^2)^(1/2)))^(1/2)*Elli 
pticF(1/2*x*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))/(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)

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c})}{2 \, c} \] Input:

integrate(1/(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")
 

Output:

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

Sympy [F]

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

integrate(1/(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)
 

Output:

Integral(1/(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]

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

integrate(1/(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")
 

Output:

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

Giac [F]

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

integrate(1/(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")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx=\frac {\sqrt {\sqrt {4 a c +b^{2}}+b}\, \sqrt {\sqrt {4 a c +b^{2}}-b}\, \left (\int \frac {\sqrt {\sqrt {4 a c +b^{2}}+b +2 c \,x^{2}}\, \sqrt {\sqrt {4 a c +b^{2}}-b -2 c \,x^{2}}}{-c \,x^{4}-b \,x^{2}+a}d x \right )}{4 c} \] Input:

int(1/(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)
 

Output:

(sqrt(sqrt(4*a*c + b**2) + b)*sqrt(sqrt(4*a*c + b**2) - b)*int((sqrt(sqrt( 
4*a*c + b**2) + b + 2*c*x**2)*sqrt(sqrt(4*a*c + b**2) - b - 2*c*x**2))/(a 
- b*x**2 - c*x**4),x))/(4*c)