3.1896 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x^4} \, dx\)

Optimal. Leaf size=74 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 b^{3/2}}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3} \]

[Out]

-Sqrt[a + b/x^2]/(4*x^3) - (a*Sqrt[a + b/x^2])/(8*b*x) + (a^2*ArcTanh[Sqrt[b]/(S
qrt[a + b/x^2]*x)])/(8*b^(3/2))

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Rubi [A]  time = 0.109531, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 b^{3/2}}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]/x^4,x]

[Out]

-Sqrt[a + b/x^2]/(4*x^3) - (a*Sqrt[a + b/x^2])/(8*b*x) + (a^2*ArcTanh[Sqrt[b]/(S
qrt[a + b/x^2]*x)])/(8*b^(3/2))

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Rubi in Sympy [A]  time = 10.5902, size = 60, normalized size = 0.81 \[ \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{8 b^{\frac{3}{2}}} - \frac{a \sqrt{a + \frac{b}{x^{2}}}}{8 b x} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{4 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**(1/2)/x**4,x)

[Out]

a**2*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(8*b**(3/2)) - a*sqrt(a + b/x**2)/(8*b*
x) - sqrt(a + b/x**2)/(4*x**3)

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Mathematica [A]  time = 0.0958387, size = 98, normalized size = 1.32 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (-a^2 x^4 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+a^2 x^4 \log (x)+\sqrt{b} \sqrt{a x^2+b} \left (a x^2+2 b\right )\right )}{8 b^{3/2} x^3 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]/x^4,x]

[Out]

-(Sqrt[a + b/x^2]*(Sqrt[b]*Sqrt[b + a*x^2]*(2*b + a*x^2) + a^2*x^4*Log[x] - a^2*
x^4*Log[b + Sqrt[b]*Sqrt[b + a*x^2]]))/(8*b^(3/2)*x^3*Sqrt[b + a*x^2])

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Maple [A]  time = 0.012, size = 106, normalized size = 1.4 \[{\frac{1}{8\,{b}^{2}{x}^{3}}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{4}{a}^{2}-\sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}+ \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{2}a-2\, \left ( a{x}^{2}+b \right ) ^{3/2}b \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)/x^4,x)

[Out]

1/8*((a*x^2+b)/x^2)^(1/2)/x^3*(b^(1/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^4*a
^2-(a*x^2+b)^(1/2)*x^4*a^2+(a*x^2+b)^(3/2)*x^2*a-2*(a*x^2+b)^(3/2)*b)/(a*x^2+b)^
(1/2)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261494, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \sqrt{b} x^{3} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, b^{2} x^{3}}, -\frac{a^{2} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)/x^4,x, algorithm="fricas")

[Out]

[1/16*(a^2*sqrt(b)*x^3*log(-(2*b*x*sqrt((a*x^2 + b)/x^2) + (a*x^2 + 2*b)*sqrt(b)
)/x^2) - 2*(a*b*x^2 + 2*b^2)*sqrt((a*x^2 + b)/x^2))/(b^2*x^3), -1/8*(a^2*sqrt(-b
)*x^3*arctan(sqrt(-b)/(x*sqrt((a*x^2 + b)/x^2))) + (a*b*x^2 + 2*b^2)*sqrt((a*x^2
+ b)/x^2))/(b^2*x^3)]

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Sympy [A]  time = 12.8142, size = 92, normalized size = 1.24 \[ - \frac{a^{\frac{3}{2}}}{8 b x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{3 \sqrt{a}}{8 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8 b^{\frac{3}{2}}} - \frac{b}{4 \sqrt{a} x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(1/2)/x**4,x)

[Out]

-a**(3/2)/(8*b*x*sqrt(1 + b/(a*x**2))) - 3*sqrt(a)/(8*x**3*sqrt(1 + b/(a*x**2)))
+ a**2*asinh(sqrt(b)/(sqrt(a)*x))/(8*b**(3/2)) - b/(4*sqrt(a)*x**5*sqrt(1 + b/(
a*x**2)))

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GIAC/XCAS [A]  time = 0.24671, size = 86, normalized size = 1.16 \[ -\frac{1}{8} \, a^{2}{\left (\frac{\arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (a x^{2} + b\right )}^{\frac{3}{2}} + \sqrt{a x^{2} + b} b}{a^{2} b x^{4}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^2)/x^4,x, algorithm="giac")

[Out]

-1/8*a^2*(arctan(sqrt(a*x^2 + b)/sqrt(-b))/(sqrt(-b)*b) + ((a*x^2 + b)^(3/2) + s
qrt(a*x^2 + b)*b)/(a^2*b*x^4))*sign(x)