3.1693 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x} \, dx\)

Optimal. Leaf size=39 \[ 2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 \sqrt{a+\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.0664272, antiderivative size = 39, 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 \[ 2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.73073, size = 31, normalized size = 0.79 \[ 2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 2 \sqrt{a + \frac{b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0259487, size = 46, normalized size = 1.18 \[ \sqrt{a} \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )-2 \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]/x,x]

[Out]

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

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Maple [B]  time = 0.01, size = 103, normalized size = 2.6 \[ -{\frac{1}{bx}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}ab+2\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238334, size = 1, normalized size = 0.03 \[ \left [\sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \, \sqrt{\frac{a x + b}{x}}, 2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) - 2 \, \sqrt{\frac{a x + b}{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.94588, size = 68, normalized size = 1.74 \[ 2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{2 a \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{2 \sqrt{b}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(1/2)/x,x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError