3.172 \(\int \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{2 d \sqrt{a+\frac{b}{x^2}}}{x \sqrt{c+\frac{d}{x^2}}}+x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}-\frac{\sqrt{c} \sqrt{a+\frac{b}{x^2}} (a d+b c) F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{2 \sqrt{c} \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

[Out]

(-2*d*Sqrt[a + b/x^2])/(Sqrt[c + d/x^2]*x) + Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x +
(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b/x^2]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (
b*c)/(a*d)])/(Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2]) - (Sqrt[c]*
(b*c + a*d)*Sqrt[a + b/x^2]*EllipticF[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*
d)])/(a*Sqrt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.633792, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{2 d \sqrt{a+\frac{b}{x^2}}}{x \sqrt{c+\frac{d}{x^2}}}+x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}-\frac{\sqrt{c} \sqrt{a+\frac{b}{x^2}} (a d+b c) F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{2 \sqrt{c} \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]*Sqrt[c + d/x^2],x]

[Out]

(-2*d*Sqrt[a + b/x^2])/(Sqrt[c + d/x^2]*x) + Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x +
(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b/x^2]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (
b*c)/(a*d)])/(Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2]) - (Sqrt[c]*
(b*c + a*d)*Sqrt[a + b/x^2]*EllipticF[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*
d)])/(a*Sqrt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 62.9457, size = 202, normalized size = 0.87 \[ - \frac{\sqrt{a} \sqrt{c + \frac{d}{x^{2}}} \left (a d + b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{b} c \sqrt{\frac{a \left (c + \frac{d}{x^{2}}\right )}{c \left (a + \frac{b}{x^{2}}\right )}} \sqrt{a + \frac{b}{x^{2}}}} + \frac{2 \sqrt{c} \sqrt{d} \sqrt{a + \frac{b}{x^{2}}} E\left (\operatorname{atan}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}\middle | 1 - \frac{b c}{a d}\right )}{\sqrt{\frac{c \left (a + \frac{b}{x^{2}}\right )}{a \left (c + \frac{d}{x^{2}}\right )}} \sqrt{c + \frac{d}{x^{2}}}} - \frac{2 d \sqrt{a + \frac{b}{x^{2}}}}{x \sqrt{c + \frac{d}{x^{2}}}} + x \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d/x**2)**(1/2)*(a+b/x**2)**(1/2),x)

[Out]

-sqrt(a)*sqrt(c + d/x**2)*(a*d + b*c)*elliptic_f(atan(sqrt(b)/(sqrt(a)*x)), -a*d
/(b*c) + 1)/(sqrt(b)*c*sqrt(a*(c + d/x**2)/(c*(a + b/x**2)))*sqrt(a + b/x**2)) +
2*sqrt(c)*sqrt(d)*sqrt(a + b/x**2)*elliptic_e(atan(sqrt(d)/(sqrt(c)*x)), 1 - b*
c/(a*d))/(sqrt(c*(a + b/x**2)/(a*(c + d/x**2)))*sqrt(c + d/x**2)) - 2*d*sqrt(a +
b/x**2)/(x*sqrt(c + d/x**2)) + x*sqrt(a + b/x**2)*sqrt(c + d/x**2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.519844, size = 205, normalized size = 0.88 \[ -\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} \left (\sqrt{\frac{a}{b}} \left (a x^2+b\right ) \left (c x^2+d\right )+i x \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} (b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{\frac{a}{b}} x\right )|\frac{b c}{a d}\right )+2 i a d x \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a}{b}} x\right )|\frac{b c}{a d}\right )\right )}{\sqrt{\frac{a}{b}} \left (a x^2+b\right ) \left (c x^2+d\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]*Sqrt[c + d/x^2],x]

[Out]

-((Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x*(Sqrt[a/b]*(b + a*x^2)*(d + c*x^2) + (2*I)*
a*d*x*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticE[I*ArcSinh[Sqrt[a/b]*x],
(b*c)/(a*d)] + I*(b*c - a*d)*x*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticF
[I*ArcSinh[Sqrt[a/b]*x], (b*c)/(a*d)]))/(Sqrt[a/b]*(b + a*x^2)*(d + c*x^2)))

_______________________________________________________________________________________

Maple [A]  time = 0.05, size = 277, normalized size = 1.2 \[{\frac{x}{ac{x}^{4}+ad{x}^{2}+bc{x}^{2}+bd}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( -\sqrt{-{\frac{c}{d}}}{x}^{4}ac+\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}{\it EllipticF} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) xad-cb\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}x{\it EllipticF} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) +2\,cb\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}x{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) -\sqrt{-{\frac{c}{d}}}{x}^{2}ad-\sqrt{-{\frac{c}{d}}}{x}^{2}bc-\sqrt{-{\frac{c}{d}}}bd \right ){\frac{1}{\sqrt{-{\frac{c}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x^2)^(1/2)*(a+b/x^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)*sqrt(c + d/x^2),x, algorithm="maxima")

[Out]

integrate(sqrt(a + b/x^2)*sqrt(c + d/x^2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\frac{a x^{2} + b}{x^{2}}} \sqrt{\frac{c x^{2} + d}{x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)*sqrt(c + d/x^2),x, algorithm="fricas")

[Out]

integral(sqrt((a*x^2 + b)/x^2)*sqrt((c*x^2 + d)/x^2), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d/x**2)**(1/2)*(a+b/x**2)**(1/2),x)

[Out]

Integral(sqrt(a + b/x**2)*sqrt(c + d/x**2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)*sqrt(c + d/x^2),x, algorithm="giac")

[Out]

integrate(sqrt(a + b/x^2)*sqrt(c + d/x^2), x)