3.1724 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0505647, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.07538, size = 20, normalized size = 0.8 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0190876, size = 34, normalized size = 1.36 \[ \frac{\log \left (2 \sqrt{a} x \sqrt{\frac{a x+b}{x}}+2 a x+b\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*x),x]

[Out]

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

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Maple [B]  time = 0.016, size = 119, normalized size = 4.8 \[{\frac{x}{2\,b}\sqrt{{\frac{ax+b}{x}}} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) +b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) +2\,\sqrt{a{x}^{2}+bx}\sqrt{a}-2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b/x)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238194, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{\sqrt{a}}, -\frac{2 \, \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.11737, size = 22, normalized size = 0.88 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b/x)**(1/2),x)

[Out]

2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/sqrt(a)

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GIAC/XCAS [A]  time = 0.243754, size = 34, normalized size = 1.36 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x),x, algorithm="giac")

[Out]

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