\(\int \frac {1}{\sqrt {1-\frac {(b-\sqrt {b^2+4 a c}) x^2}{2 a}} \sqrt {1-\frac {(b+\sqrt {b^2+4 a c}) x^2}{2 a}}} \, dx\) [294]

Optimal result

Integrand size = 63, antiderivative size = 94 \[ \int \frac {1}{\sqrt {1-\frac {\left (b-\sqrt {b^2+4 a c}\right ) x^2}{2 a}} \sqrt {1-\frac {\left (b+\sqrt {b^2+4 a c}\right ) x^2}{2 a}}} \, dx=\frac {\sqrt {2} \sqrt {a} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x}{\sqrt {2} \sqrt {a}}\right ),\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {b+\sqrt {b^2+4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 5.51 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt {1-\frac {\left (b-\sqrt {b^2+4 a c}\right ) x^2}{2 a}} \sqrt {1-\frac {\left (b+\sqrt {b^2+4 a c}\right ) x^2}{2 a}}} \, dx=\frac {\sqrt {2} \sqrt {a} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x}{\sqrt {2} \sqrt {a}}\right ),-\frac {-b+\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {b+\sqrt {b^2+4 a c}}} \] Input:

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

Output:

(Sqrt[2]*Sqrt[a]*EllipticF[ArcSin[(Sqrt[b + Sqrt[b^2 + 4*a*c]]*x)/(Sqrt[2] 
*Sqrt[a])], -((-b + Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*c]))])/Sqrt[b + 
 Sqrt[b^2 + 4*a*c]]
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {321}

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.

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(Sqrt[2]*Sqrt[a]*EllipticF[ArcSin[(Sqrt[b + Sqrt[b^2 + 4*a*c]]*x)/(Sqrt[2] 
*Sqrt[a])], (b - Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*c])])/Sqrt[b + Sqr 
t[b^2 + 4*a*c]]
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(76)=152\).

Time = 1.45 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.81

method	result	size
elliptic	\(\frac {\sqrt {-\frac {\left (x^{2} \sqrt {4 a c +b^{2}}-b \,x^{2}+2 a \right ) \left (x^{2} \sqrt {4 a c +b^{2}}+b \,x^{2}-2 a \right )}{a^{2}}}\, \sqrt {2}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {x^{2} \sqrt {4 a c +b^{2}}-b \,x^{2}+2 a}{a}}\, \sqrt {-\frac {x^{2} \sqrt {4 a c +b^{2}}+b \,x^{2}-2 a}{a}}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-\frac {b \,x^{2}}{a}+1-\frac {c \,x^{4}}{a}}}\)	\(264\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {1-\frac {\left (b-\sqrt {b^2+4 a c}\right ) x^2}{2 a}} \sqrt {1-\frac {\left (b+\sqrt {b^2+4 a c}\right ) x^2}{2 a}}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c})}{2 \, c} \] Input:

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

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

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

Output:

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

Maxima [F]

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

integrate(4/(4-2*(b-(4*a*c+b^2)^(1/2))*x^2/a)^(1/2)/(4-2*(b+(4*a*c+b^2)^(1 
/2))*x^2/a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(4/(4-2*(b-(4*a*c+b^2)^(1/2))*x^2/a)^(1/2)/(4-2*(b+(4*a*c+b^2)^(1 
/2))*x^2/a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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