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

Optimal. Leaf size=43 \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

(Sqrt[a + b/x]*x)/a - (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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Rubi [A]  time = 0.059137, antiderivative size = 43, 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 \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + b/x],x]

[Out]

(Sqrt[a + b/x]*x)/a - (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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Rubi in Sympy [A]  time = 5.35715, size = 32, normalized size = 0.74 \[ \frac{x \sqrt{a + \frac{b}{x}}}{a} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b/x)/a - b*atanh(sqrt(a + b/x)/sqrt(a))/a**(3/2)

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Mathematica [A]  time = 0.0467834, size = 53, normalized size = 1.23 \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a + b/x],x]

[Out]

(Sqrt[a + b/x]*x)/a - (b*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2*a^(3/2))

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Maple [A]  time = 0.005, size = 70, normalized size = 1.6 \[ -{\frac{x}{2}\sqrt{{\frac{ax+b}{x}}} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) -2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(1/2),x)

[Out]

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

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247426, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}}}, \frac{\sqrt{-a} x \sqrt{\frac{a x + b}{x}} + b \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(a)*x*sqrt((a*x + b)/x) + b*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x +
b)*sqrt(a)))/a^(3/2), (sqrt(-a)*x*sqrt((a*x + b)/x) + b*arctan(a/(sqrt(-a)*sqrt(
(a*x + b)/x))))/(sqrt(-a)*a)]

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Sympy [A]  time = 7.83873, size = 44, normalized size = 1.02 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(1/2),x)

[Out]

sqrt(b)*sqrt(x)*sqrt(a*x/b + 1)/a - b*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

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GIAC/XCAS [A]  time = 0.237001, size = 96, normalized size = 2.23 \[ -\frac{b{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{2 \, a^{\frac{3}{2}}} + \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right )}{2 \, a^{\frac{3}{2}}{\rm sign}\left (x\right )} + \frac{\sqrt{a x^{2} + b x}}{a{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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