\(\int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx\) [235]

Optimal result

Integrand size = 59, antiderivative size = 94 \[ \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\sqrt {b+\sqrt {b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \] Output:

1/2*(b+(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(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 = 2.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\sqrt {b+\sqrt {b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {b+\sqrt {b^2-4 a c}}{-b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \] Input:

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]
 

Output:

(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])
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {327}

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.

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(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])])/(S 
qrt[2]*Sqrt[c])
 

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1387\) vs. \(2(76)=152\).

Time = 0.80 (sec) , antiderivative size = 1388, normalized size of antiderivative = 14.77

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

Input:

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)
 

Output:

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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (76) = 152\).

Time = 0.12 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.36 \[ \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (b^{2} x + \sqrt {b^{2} - 4 \, a c} b x + {\left (b c x + \sqrt {b^{2} - 4 \, a c} c x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}} \sqrt {\frac {c}{a}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,-\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b^{2} + 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (\sqrt {b^{2} - 4 \, a c} b x + {\left (b^{2} - 2 \, b c\right )} x + {\left (\sqrt {b^{2} - 4 \, a c} c x + {\left (b c + 2 \, c^{2}\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}} \sqrt {\frac {c}{a}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,-\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b^{2} + 2 \, a c}{2 \, a c}) + {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} \sqrt {-\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}} \sqrt {-\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}}}{4 \, c^{2} x} \] Input:

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="fricas")
 

Output:

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

Sympy [F]

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

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)
 

Output:

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]

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

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="maxima")
 

Output:

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

Giac [F]

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

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")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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)
 

Output:

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)