3.1754 \(\int \frac{\sqrt{a+\frac{b}{x}}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=49 \[ 2 \sqrt{x} \sqrt{a+\frac{b}{x}}-2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0734367, antiderivative size = 49, 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 \[ 2 \sqrt{x} \sqrt{a+\frac{b}{x}}-2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.22513, size = 41, normalized size = 0.84 \[ - 2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )} + 2 \sqrt{x} \sqrt{a + \frac{b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0627394, size = 59, normalized size = 1.2 \[ 2 \sqrt{x} \sqrt{a+\frac{b}{x}}-2 \sqrt{b} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\sqrt{b} \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]/Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.015, size = 50, normalized size = 1. \[ -2\,{\frac{\sqrt{x}}{\sqrt{ax+b}}\sqrt{{\frac{ax+b}{x}}} \left ( \sqrt{b}{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) -\sqrt{ax+b} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247281, size = 1, normalized size = 0.02 \[ \left [\sqrt{b} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, \sqrt{x} \sqrt{\frac{a x + b}{x}}, -2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) + 2 \, \sqrt{x} \sqrt{\frac{a x + b}{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.64725, size = 68, normalized size = 1.39 \[ \frac{2 \sqrt{a} \sqrt{x}}{\sqrt{1 + \frac{b}{a x}}} - 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )} + \frac{2 b}{\sqrt{a} \sqrt{x} \sqrt{1 + \frac{b}{a x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241553, size = 84, normalized size = 1.71 \[ 2 \,{\left (\frac{b \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \sqrt{a x + b} - \frac{b \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b} \sqrt{b}}{\sqrt{-b}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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