3.1666 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) \sqrt{x}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]

[Out]

(2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)

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Rubi [A]  time = 0.0454216, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*Sqrt[x]),x]

[Out]

(2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)

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Rubi in Sympy [A]  time = 8.22622, size = 36, normalized size = 0.9 \[ \frac{2 \sqrt{x}}{a} - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x)/a - 2*sqrt(b)*atan(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

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Mathematica [A]  time = 0.0208891, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)

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Maple [A]  time = 0.008, size = 32, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{a}}-2\,{\frac{b}{a\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239034, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \, \sqrt{x}}{a}, -\frac{2 \,{\left (\sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) - \sqrt{x}\right )}}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.90047, size = 92, normalized size = 2.3 \[ \begin{cases} \frac{2 \sqrt{x}}{a} + \frac{i \sqrt{b} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} - \frac{i \sqrt{b} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**(1/2),x)

[Out]

Piecewise((2*sqrt(x)/a + I*sqrt(b)*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(a**2*sqr
t(1/a)) - I*sqrt(b)*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))/(a**2*sqrt(1/a)), Ne(a, 0
)), (2*x**(3/2)/(3*b), True))

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GIAC/XCAS [A]  time = 0.220006, size = 42, normalized size = 1.05 \[ -\frac{2 \, b \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{2 \, \sqrt{x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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