3.3.100 \(\int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx\) [300]

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

[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)

________________________________________________________________________________________

Rubi [A]
time = 0.14, 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 = {434, 435, 430} \begin {gather*} \frac {\sqrt {2} b F\left (\text {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 (\text {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}}} \end {gather*}

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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 434

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 435

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

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*}

________________________________________________________________________________________

Mathematica [A]
time = 2.25, size = 102, normalized size = 0.47 \begin {gather*} \frac {\sqrt {-b-\sqrt {b^2-4 a c}} 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}} \end {gather*}

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])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(173)=346\).
time = 0.19, size = 872, normalized size = 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}}}}\, \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 (\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 )-\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 )}\)	\(872\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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)

________________________________________________________________________________________