3.3.99 \(\int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx\) [299]

3.3.99.1 Optimal result

Integrand size = 59, antiderivative size = 478 \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {x \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}-\frac {\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \]

output

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

3.3.99.2 Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=\frac {\sqrt {-b-\sqrt {b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \]

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])
 

3.3.99.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {324, 320, 388, 313}

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.

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

\(\displaystyle \frac {2 c \left (\frac {x \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{2 c \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) \int \frac {\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{\left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^{3/2}}dx}{2 c}\right )}{b-\sqrt {b^2-4 a c}}+\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {2 c \left (\frac {x \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}}{2 c \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\right )}{b-\sqrt {b^2-4 a c}}+\frac {\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),-\frac {2 \sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1}{\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\)

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

(2*c*(((b - Sqrt[b^2 - 4*a*c])*x*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c] 
)])/(2*c*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]) - ((b - Sqrt[b^2 - 4 
*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a* 
c])]*EllipticE[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])])/(2*Sqrt[2]*c^(3/2)*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])])))/(b - Sqrt[b^2 - 4*a*c]) 
 + (Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]) 
]*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (-2*S 
qrt[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]))]*S 
qrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])
 

3.3.99.3.1 Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

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]
 

rule 324

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

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

3.3.99.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1387\) vs. \(2(457)=914\).

Time = 2.69 (sec) , antiderivative size = 1388, normalized size of antiderivative = 2.90

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...
 

3.3.99.5 Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (b^{2} x + \sqrt {b^{2} - 4 \, a c} b x - {\left (b c x + \sqrt {b^{2} - 4 \, a c} c x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} \sqrt {\frac {c}{a}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (\sqrt {b^{2} - 4 \, a c} b x + {\left (b^{2} + 2 \, b c\right )} x - {\left (\sqrt {b^{2} - 4 \, a c} c x + {\left (b c - 2 \, c^{2}\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} \sqrt {\frac {c}{a}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} \sqrt {\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}} \sqrt {\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}}}{4 \, c^{2} x} \]

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 + s 
qrt(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)
 

3.3.99.6 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)
 

3.3.99.7 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 + sqrt 
(b^2 - 4*a*c)) + 1), x)
 

3.3.99.8 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:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con 
st gen &
 

3.3.99.9 Mupad [F(-1)]

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

input

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

output

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