3.1691 \(\int \sqrt{a+\frac{b}{x}} x \, dx\)

Optimal. Leaf size=69 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{1}{2} x^2 \sqrt{a+\frac{b}{x}}+\frac{b x \sqrt{a+\frac{b}{x}}}{4 a} \]

[Out]

(b*Sqrt[a + b/x]*x)/(4*a) + (Sqrt[a + b/x]*x^2)/2 - (b^2*ArcTanh[Sqrt[a + b/x]/S
qrt[a]])/(4*a^(3/2))

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Rubi [A]  time = 0.0903853, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{1}{2} x^2 \sqrt{a+\frac{b}{x}}+\frac{b x \sqrt{a+\frac{b}{x}}}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*x,x]

[Out]

(b*Sqrt[a + b/x]*x)/(4*a) + (Sqrt[a + b/x]*x^2)/2 - (b^2*ArcTanh[Sqrt[a + b/x]/S
qrt[a]])/(4*a^(3/2))

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Rubi in Sympy [A]  time = 8.8529, size = 53, normalized size = 0.77 \[ \frac{x^{2} \sqrt{a + \frac{b}{x}}}{2} + \frac{b x \sqrt{a + \frac{b}{x}}}{4 a} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b/x)/2 + b*x*sqrt(a + b/x)/(4*a) - b**2*atanh(sqrt(a + b/x)/sqrt(a
))/(4*a**(3/2))

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Mathematica [A]  time = 0.0963773, size = 64, normalized size = 0.93 \[ \frac{x \sqrt{a+\frac{b}{x}} (2 a x+b)}{4 a}-\frac{b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]*x,x]

[Out]

(Sqrt[a + b/x]*x*(b + 2*a*x))/(4*a) - (b^2*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/
x]*x])/(8*a^(3/2))

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Maple [A]  time = 0.01, size = 96, normalized size = 1.4 \[{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( 4\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x+2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}b-{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b/x)^(1/2),x)

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(4*(a*x^2+b*x)^(1/2)*a^(5/2)*x+2*(a*x^2+b*x)^(1/2)*a^(3/
2)*b-b^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a)/(x*(a*x+b))^(1
/2)/a^(5/2)

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241606, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a x^{2} + b x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{8 \, a^{\frac{3}{2}}}, \frac{b^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (2 \, a x^{2} + b x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(b^2*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(2*a*x^2 + b*x
)*sqrt(a)*sqrt((a*x + b)/x))/a^(3/2), 1/4*(b^2*arctan(a/(sqrt(-a)*sqrt((a*x + b)
/x))) + (2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*a)]

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Sympy [A]  time = 12.1334, size = 97, normalized size = 1.41 \[ \frac{a x^{\frac{5}{2}}}{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} x^{\frac{3}{2}}}{4 \sqrt{\frac{a x}{b} + 1}} + \frac{b^{\frac{3}{2}} \sqrt{x}}{4 a \sqrt{\frac{a x}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b/x)**(1/2),x)

[Out]

a*x**(5/2)/(2*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*x**(3/2)/(4*sqrt(a*x/b + 1))
+ b**(3/2)*sqrt(x)/(4*a*sqrt(a*x/b + 1)) - b**2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(
4*a**(3/2))

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GIAC/XCAS [A]  time = 0.244618, size = 105, normalized size = 1.52 \[ -\frac{b^{2}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{8 \, a^{\frac{3}{2}}} + \frac{1}{8} \,{\left (2 \, \sqrt{a x^{2} + b x}{\left (2 \, x + \frac{b}{a}\right )} + \frac{b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right )}{a^{\frac{3}{2}}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*b^2*ln(abs(b))*sign(x)/a^(3/2) + 1/8*(2*sqrt(a*x^2 + b*x)*(2*x + b/a) + b^2
*ln(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))/a^(3/2))*sign(x)