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

Optimal. Leaf size=232 \[ -\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

[Out]

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

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Rubi [A]  time = 0.587952, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 56.6809, size = 199, normalized size = 0.86 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{c + \frac{d}{x^{2}}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}\middle | - \frac{a d}{b c} + 1\right )}{c \sqrt{\frac{a \left (c + \frac{d}{x^{2}}\right )}{c \left (a + \frac{b}{x^{2}}\right )}} \sqrt{a + \frac{b}{x^{2}}}} - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{c x \sqrt{a + \frac{b}{x^{2}}}} + \frac{x \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}}}{c} - \frac{b \sqrt{c} \sqrt{a + \frac{b}{x^{2}}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}\middle | 1 - \frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{\frac{c \left (a + \frac{b}{x^{2}}\right )}{a \left (c + \frac{d}{x^{2}}\right )}} \sqrt{c + \frac{d}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101855, size = 86, normalized size = 0.37 \[ \frac{\sqrt{a+\frac{b}{x^2}} \sqrt{\frac{c x^2+d}{d}} E\left (\sin ^{-1}\left (\sqrt{-\frac{c}{d}} x\right )|\frac{a d}{b c}\right )}{\sqrt{-\frac{c}{d}} \sqrt{\frac{a x^2+b}{b}} \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.029, size = 94, normalized size = 0.4 \[{\frac{b}{a{x}^{2}+b}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) \sqrt{{\frac{a{x}^{2}+b}{b}}}\sqrt{{\frac{c{x}^{2}+d}{d}}}{\frac{1}{\sqrt{-{\frac{c}{d}}}}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)