3.1892 \(\int \sqrt{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=42 \[ x \sqrt{a+\frac{b}{x^2}}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0617865, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ x \sqrt{a+\frac{b}{x^2}}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.38026, size = 34, normalized size = 0.81 \[ - \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )} + x \sqrt{a + \frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0504626, size = 71, normalized size = 1.69 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b}-\sqrt{b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+\sqrt{b} \log (x)\right )}{\sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2],x]

[Out]

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

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Maple [A]  time = 0.008, size = 63, normalized size = 1.5 \[ -{x\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) -\sqrt{a{x}^{2}+b} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2),x)

[Out]

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

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251483, size = 1, normalized size = 0.02 \[ \left [x \sqrt{\frac{a x^{2} + b}{x^{2}}} + \frac{1}{2} \, \sqrt{b} \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ), x \sqrt{\frac{a x^{2} + b}{x^{2}}} - \sqrt{-b} \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.27006, size = 56, normalized size = 1.33 \[ \frac{\sqrt{a} x}{\sqrt{1 + \frac{b}{a x^{2}}}} - \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )} + \frac{b}{\sqrt{a} x \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.233782, size = 92, normalized size = 2.19 \[{\left (\frac{b \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \sqrt{a x^{2} + b}\right )}{\rm sign}\left (x\right ) - \frac{{\left (b \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b} \sqrt{b}\right )}{\rm sign}\left (x\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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