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

Optimal. Leaf size=215 \[ \frac {\sqrt {2} b \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ),-\frac {b-\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\sqrt {b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}} \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {423, 424, 419} \[ \frac {\sqrt {2} b F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\sqrt {b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|-\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}} \]

Antiderivative was successfully verified.

[In]

Int[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]

[Out]

-(((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]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*b*EllipticF[Arc
Sin[(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]))])/(Sq
rt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), 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 423

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[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 424

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

Rubi steps

\begin {align*} \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 {(2 b) \int \frac {1}{\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}{b-\sqrt {b^2-4 a c}}-\frac {\left (b+\sqrt {b^2-4 a c}\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}{b-\sqrt {b^2-4 a c}}\\ &=-\frac {\left (b+\sqrt {b^2-4 a c}\right ) E\left (\sin ^{-1}\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 F\left (\sin ^{-1}\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}}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 102, normalized size = 0.47 \[ \frac {\sqrt {-\sqrt {b^2-4 a c}-b} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}-b}\right )}{\sqrt {2} \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

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]

[Out]

(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])

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a\right )} \sqrt {-\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}} \sqrt {\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}}}{4 \, {\left (c x^{4} + b x^{2} + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="fric
as")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, choosing root of [1,0,%%%{8,[1,0,
1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[
1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [71.707969239,70,22]Warning, choosing ro
ot of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{
1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [78.6493344628,0,0
]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%
%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters valu
es [50.5901726987,49,-6]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%}
,0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%
%%}] at parameters values [91.0141688026,0,0]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%
%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,
0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [94.3029591851,0,0]Warning, choosing root of [1,0,%%%{8,[1,1,0]
%%%}+%%%{-4,[1,0,0]%%%}+%%%{-2,[0,0,2]%%%},0,%%%{16,[2,2,0]%%%}+%%%{16,[2,1,0]%%%}+%%%{4,[2,0,0]%%%}+%%%{-8,[1
,1,2]%%%}+%%%{-4,[1,0,2]%%%}+%%%{1,[0,0,4]%%%}] at parameters values [-64,2,62]Evaluation time: 28.88

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}+1}}{\sqrt {\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-\frac {2 \, c x^{2}}{b - \sqrt {b^{2} - 4 \, a c}} + 1}}{\sqrt {\frac {2 \, c x^{2}}{b + \sqrt {b^{2} - 4 \, a c}} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="maxi
ma")

[Out]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {1-\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}}}{\sqrt {\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {- b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{b - \sqrt {- 4 a c + b^{2}}}}}{\sqrt {\frac {b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{b + \sqrt {- 4 a c + b^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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