3.1894 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 \sqrt{b}} \]

[Out]

-Sqrt[a + b/x^2]/(2*x) - (a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(2*Sqrt[b])

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Rubi [A]  time = 0.0626728, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]/x^2,x]

[Out]

-Sqrt[a + b/x^2]/(2*x) - (a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(2*Sqrt[b])

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Rubi in Sympy [A]  time = 5.01322, size = 41, normalized size = 0.82 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2 \sqrt{b}} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0560553, size = 86, normalized size = 1.72 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-\sqrt{b} \sqrt{a x^2+b}-a x^2 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+a x^2 \log (x)\right )}{2 \sqrt{b} x \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 85, normalized size = 1.7 \[ -{\frac{1}{2\,bx}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}a-\sqrt{a{x}^{2}+b}{x}^{2}a+ \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)/x^2,x)

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24781, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{b} x \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \, b x}, \frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) - b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \, b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(a*sqrt(b)*x*log((2*b*x*sqrt((a*x^2 + b)/x^2) - (a*x^2 + 2*b)*sqrt(b))/x^2)
- 2*b*sqrt((a*x^2 + b)/x^2))/(b*x), 1/2*(a*sqrt(-b)*x*arctan(sqrt(-b)/(x*sqrt((
a*x^2 + b)/x^2))) - b*sqrt((a*x^2 + b)/x^2))/(b*x)]

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Sympy [A]  time = 6.92504, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{2}}}}{2 x} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*sqrt(1 + b/(a*x**2))/(2*x) - a*asinh(sqrt(b)/(sqrt(a)*x))/(2*sqrt(b))

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GIAC/XCAS [A]  time = 0.248393, size = 61, normalized size = 1.22 \[ \frac{1}{2} \, a{\left (\frac{\arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{a x^{2} + b}}{a x^{2}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*(arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) - sqrt(a*x^2 + b)/(a*x^2))*sign
(x)