\(\int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx\) [237]

Optimal result

Integrand size = 59, antiderivative size = 215 \[ \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=-\frac {\left (b+\sqrt {b^2-4 a c}\right ) E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\sqrt {-b-\sqrt {b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{-b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \] 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 = 0.38 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {326, 321, 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.

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

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:

-(((b + Sqrt[b^2 - 4*a*c])*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - S 
qrt[b^2 - 4*a*c]]], -((b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c]))])/( 
Sqrt[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*b*EllipticF[ArcSi 
n[(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[c]*Sqrt[b - Sqrt[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])
 

rule 326

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

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. \(871\) vs. \(2(175)=350\).

Time = 0.64 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.06

method	result	size
elliptic	\(\frac {\sqrt {\frac {2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}}\, \left (-b +\sqrt {-4 a c +b^{2}}\right ) \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}}\, \left (\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}}}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4-\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 (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-b +\sqrt {-4 a c +b^{2}}\right )}}}{2}\right )}{\sqrt {-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}}\, \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 )}}}-\frac {4 c \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}}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4-\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 (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-b +\sqrt {-4 a c +b^{2}}\right )}}}{2}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4-\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 (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-b +\sqrt {-4 a c +b^{2}}\right )}}}{2}\right )\right )}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}}\, \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 )}}\, \left (\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}-\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}-\frac {b}{a}\right )}\right )}{2 \sqrt {\frac {2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}\)	\(872\)

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {4 \, \sqrt {\frac {1}{2}} a x \sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}} \sqrt {-\frac {c}{a}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}{x}\right )\,|\,-\frac {b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} b}{2 \, a c}) - 2 \, \sqrt {\frac {1}{2}} {\left ({\left (2 \, a - b\right )} x - \sqrt {b^{2} - 4 \, a c} x\right )} \sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}} \sqrt {-\frac {c}{a}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}{x}\right )\,|\,-\frac {b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} b}{2 \, a c}) + {\left (b + \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {-\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}} \sqrt {\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}}}{4 \, c x} \] 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*(4*sqrt(1/2)*a*x*sqrt((b - sqrt(b^2 - 4*a*c))/c)*sqrt(-c/a)*elliptic_e 
(arcsin(sqrt(1/2)*sqrt((b - sqrt(b^2 - 4*a*c))/c)/x), -1/2*(b^2 - 2*a*c + 
sqrt(b^2 - 4*a*c)*b)/(a*c)) - 2*sqrt(1/2)*((2*a - b)*x - sqrt(b^2 - 4*a*c) 
*x)*sqrt((b - sqrt(b^2 - 4*a*c))/c)*sqrt(-c/a)*elliptic_f(arcsin(sqrt(1/2) 
*sqrt((b - sqrt(b^2 - 4*a*c))/c)/x), -1/2*(b^2 - 2*a*c + sqrt(b^2 - 4*a*c) 
*b)/(a*c)) + (b + sqrt(b^2 - 4*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)
 

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

Giac [F(-2)]

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

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.Non regula 
r value [
 

Mupad [F(-1)]

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

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

Output:

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