3.1780 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0454773, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.37622, size = 27, normalized size = 0.9 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(1/2)/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0276437, size = 36, normalized size = 1.2 \[ \frac{\log (x)-2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*x^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 39, normalized size = 1.3 \[ -2\,{\frac{\sqrt{x}}{\sqrt{ax+b}\sqrt{b}}\sqrt{{\frac{ax+b}{x}}}{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(1/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2407, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right )}{\sqrt{b}}, \frac{2 \, \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 25.1049, size = 24, normalized size = 0.8 \[ - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(1/2)/x**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.230429, size = 53, normalized size = 1.77 \[ \frac{2 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(3/2)),x, algorithm="giac")

[Out]

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