3.1755 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^{3/2}} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{x}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0645598, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{x}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.437, size = 41, normalized size = 0.82 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{\sqrt{b}} - \frac{\sqrt{a + \frac{b}{x}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0958227, size = 64, normalized size = 1.28 \[ -\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{x}}-\frac{a \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )}{\sqrt{b}}+\frac{a \log (x)}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 54, normalized size = 1.1 \[ -{1\sqrt{{\frac{ax+b}{x}}} \left ({\it Artanh} \left ({1\sqrt{ax+b}{\frac{1}{\sqrt{b}}}} \right ) ax+\sqrt{ax+b}\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251096, size = 1, normalized size = 0.02 \[ \left [\frac{a x \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \, \sqrt{b} x}, \frac{a x \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) - \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 19.0155, size = 44, normalized size = 0.88 \[ - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x}}}{\sqrt{x}} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28006, size = 54, normalized size = 1.08 \[ a{\left (\frac{\arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{a x + b}}{a x}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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