3.174 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{\left (c+\frac{d}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=262 \[ \frac{2 \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 x \sqrt{c+\frac{d}{x^2}}}+\frac{2 x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c^2}-\frac{x \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c} \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

[Out]

(-2*d*Sqrt[a + b/x^2])/(c^2*Sqrt[c + d/x^2]*x) - (Sqrt[a + b/x^2]*x)/(c*Sqrt[c +
d/x^2]) + (2*Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x)/c^2 + (2*Sqrt[d]*Sqrt[a + b/x^2
]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(c^(3/2)*Sqrt[(c*(a +
b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2]) - (b*Sqrt[a + b/x^2]*EllipticF[ArcCot
[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*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 = 0.835678, antiderivative size = 262, 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 \[ \frac{2 \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 x \sqrt{c+\frac{d}{x^2}}}+\frac{2 x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c^2}-\frac{x \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c} \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]/(c + d/x^2)^(3/2),x]

[Out]

(-2*d*Sqrt[a + b/x^2])/(c^2*Sqrt[c + d/x^2]*x) - (Sqrt[a + b/x^2]*x)/(c*Sqrt[c +
d/x^2]) + (2*Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x)/c^2 + (2*Sqrt[d]*Sqrt[a + b/x^2
]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(c^(3/2)*Sqrt[(c*(a +
b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2]) - (b*Sqrt[a + b/x^2]*EllipticF[ArcCot
[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*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 = 80.5002, size = 230, normalized size = 0.88 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{c + \frac{d}{x^{2}}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}\middle | - \frac{a d}{b c} + 1\right )}{c^{2} \sqrt{\frac{a \left (c + \frac{d}{x^{2}}\right )}{c \left (a + \frac{b}{x^{2}}\right )}} \sqrt{a + \frac{b}{x^{2}}}} - \frac{x \sqrt{a + \frac{b}{x^{2}}}}{c \sqrt{c + \frac{d}{x^{2}}}} - \frac{2 d \sqrt{a + \frac{b}{x^{2}}}}{c^{2} x \sqrt{c + \frac{d}{x^{2}}}} + \frac{2 x \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}}}{c^{2}} + \frac{2 \sqrt{d} \sqrt{a + \frac{b}{x^{2}}} E\left (\operatorname{atan}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}\middle | 1 - \frac{b c}{a d}\right )}{c^{\frac{3}{2}} \sqrt{\frac{c \left (a + \frac{b}{x^{2}}\right )}{a \left (c + \frac{d}{x^{2}}\right )}} \sqrt{c + \frac{d}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**(1/2)/(c+d/x**2)**(3/2),x)

[Out]

-sqrt(a)*sqrt(b)*sqrt(c + d/x**2)*elliptic_f(atan(sqrt(b)/(sqrt(a)*x)), -a*d/(b*
c) + 1)/(c**2*sqrt(a*(c + d/x**2)/(c*(a + b/x**2)))*sqrt(a + b/x**2)) - x*sqrt(a
+ b/x**2)/(c*sqrt(c + d/x**2)) - 2*d*sqrt(a + b/x**2)/(c**2*x*sqrt(c + d/x**2))
+ 2*x*sqrt(a + b/x**2)*sqrt(c + d/x**2)/c**2 + 2*sqrt(d)*sqrt(a + b/x**2)*ellip
tic_e(atan(sqrt(d)/(sqrt(c)*x)), 1 - b*c/(a*d))/(c**(3/2)*sqrt(c*(a + b/x**2)/(a
*(c + d/x**2)))*sqrt(c + d/x**2))

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Mathematica [C]  time = 0.376759, size = 191, normalized size = 0.73 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (i \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} (b c-2 a d) F\left (i \sinh ^{-1}\left (\sqrt{\frac{a}{b}} x\right )|\frac{b c}{a d}\right )+2 i a d \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a}{b}} x\right )|\frac{b c}{a d}\right )+c x \sqrt{\frac{a}{b}} \left (a x^2+b\right )\right )}{c^2 \sqrt{\frac{a}{b}} \left (a x^2+b\right ) \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]/(c + d/x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.065, size = 185, normalized size = 0.7 \[ -{\frac{c{x}^{2}+d}{{x}^{2}c \left ( a{x}^{2}+b \right ) }\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ({x}^{3}a\sqrt{-{\frac{c}{d}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) b\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}-2\,{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) b\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}+xb\sqrt{-{\frac{c}{d}}} \right ){\frac{1}{\sqrt{-{\frac{c}{d}}}}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)/(c+d/x^2)^(3/2),x)

[Out]

-((a*x^2+b)/x^2)^(1/2)/x^2/(a*x^2+b)*(x^3*a*(-c/d)^(1/2)+EllipticF(x*(-c/d)^(1/2
),(a*d/b/c)^(1/2))*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)-2*EllipticE(x*(-c/d
)^(1/2),(a*d/b/c)^(1/2))*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)+x*b*(-c/d)^(1
/2))*(c*x^2+d)/(-c/d)^(1/2)/c/((c*x^2+d)/x^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(1/2)/(c+d/x**2)**(3/2),x)

[Out]

Integral(sqrt(a + b/x**2)/(c + d/x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(a + b/x^2)/(c + d/x^2)^(3/2), x)