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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0574037, antiderivative size = 39, 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 \[ x \sqrt{a+\frac{b}{x}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.90255, size = 31, normalized size = 0.79 \[ x \sqrt{a + \frac{b}{x}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0350874, size = 50, normalized size = 1.28 \[ x \sqrt{a+\frac{b}{x}}+\frac{b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x],x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.006, size = 74, normalized size = 1.9 \[{\frac{x}{2}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.257149, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, \sqrt{a}}, \frac{\sqrt{-a} x \sqrt{\frac{a x + b}{x}} - b \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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)))/sqrt(a), (sqrt(-a)*x*sqrt((a*x + b)/x) - b*arctan(a/(sqrt(-a)*sqrt((
a*x + b)/x))))/sqrt(-a)]

_______________________________________________________________________________________

Sympy [A]  time = 6.49426, size = 42, normalized size = 1.08 \[ \sqrt{b} \sqrt{x} \sqrt{\frac{a x}{b} + 1} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232805, size = 86, normalized size = 2.21 \[ -\frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{2 \, \sqrt{a}} + \frac{b{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{2 \, \sqrt{a}} + \sqrt{a x^{2} + b x}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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