3.1893 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x} \, dx\)

Optimal. Leaf size=38 \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-\sqrt{a+\frac{b}{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0765745, antiderivative size = 38, 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 \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-\sqrt{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.66849, size = 31, normalized size = 0.82 \[ \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )} - \sqrt{a + \frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0406865, size = 54, normalized size = 1.42 \[ \sqrt{a+\frac{b}{x^2}} \left (\frac{\sqrt{a} x \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a x^2+b}}-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]/x,x]

[Out]

Sqrt[a + b/x^2]*(-1 + (Sqrt[a]*x*Log[a*x + Sqrt[a]*Sqrt[b + a*x^2]])/Sqrt[b + a*
x^2])

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Maple [B]  time = 0.01, size = 81, normalized size = 2.1 \[{\frac{1}{b}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ({a}^{{\frac{3}{2}}}\sqrt{a{x}^{2}+b}{x}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}\sqrt{a}+\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) xab \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)/x,x)

[Out]

((a*x^2+b)/x^2)^(1/2)*(a^(3/2)*(a*x^2+b)^(1/2)*x^2-(a*x^2+b)^(3/2)*a^(1/2)+ln(a^
(1/2)*x+(a*x^2+b)^(1/2))*x*a*b)/(a*x^2+b)^(1/2)/b/a^(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248816, size = 1, normalized size = 0.03 \[ \left [\frac{1}{2} \, \sqrt{a} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) - \sqrt{\frac{a x^{2} + b}{x^{2}}}, \sqrt{-a} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) - \sqrt{\frac{a x^{2} + b}{x^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.23451, size = 56, normalized size = 1.47 \[ \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )} - \frac{a x}{\sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{\sqrt{b}}{x \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.247001, size = 82, normalized size = 2.16 \[ -\frac{1}{2} \, \sqrt{a}{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \, \sqrt{a} b{\rm sign}\left (x\right )}{{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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