\(\int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx\) [50]

Optimal result

Integrand size = 87, antiderivative size = 113 \[ \int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx=-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92 \[ \int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx=-2 i \sqrt {2} a \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} x\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right ) \] Input:

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

Output:

(-2*I)*Sqrt[2]*a*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt 
[2]*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt 
[b^2 - 4*a*c])]
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {281, 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.

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 327

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

Input:

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

Defintions of rubi rules used

rule 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2539\) vs. \(2(92)=184\).

Time = 1.41 (sec) , antiderivative size = 2540, normalized size of antiderivative = 22.48

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2540\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (92) = 184\).

Time = 0.12 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.27 \[ \int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left (a c x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + a b x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}} \sqrt {\frac {c}{a}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,-\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b^{2} + 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left (\sqrt {b^{2} - 4 \, a c} b x - {\left (2 \, a b - b^{2}\right )} x - {\left ({\left (2 \, a + b\right )} c x + \sqrt {b^{2} - 4 \, a c} c x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}} \sqrt {\frac {c}{a}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,-\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b^{2} + 2 \, a c}{2 \, a c}) + 2 \, a c \sqrt {-\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}} \sqrt {-\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}}}{2 \, c x} \] Input:

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

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

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

Maxima [F]

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

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

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

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

Mupad [F(-1)]

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

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

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

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