3.1665 \(\int \frac{\sqrt{x}}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a} \]

[Out]

(-2*b*Sqrt[x])/a^2 + (2*x^(3/2))/(3*a) + (2*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqr
t[b]])/a^(5/2)

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Rubi [A]  time = 0.0602211, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + b/x),x]

[Out]

(-2*b*Sqrt[x])/a^2 + (2*x^(3/2))/(3*a) + (2*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqr
t[b]])/a^(5/2)

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Rubi in Sympy [A]  time = 10.8104, size = 49, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}}}{3 a} - \frac{2 b \sqrt{x}}{a^{2}} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/(3*a) - 2*b*sqrt(x)/a**2 + 2*b**(3/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/a
**(5/2)

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Mathematica [A]  time = 0.0368486, size = 49, normalized size = 0.92 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}+\frac{2 \sqrt{x} (a x-3 b)}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(a + b/x),x]

[Out]

(2*Sqrt[x]*(-3*b + a*x))/(3*a^2) + (2*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])
/a^(5/2)

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Maple [A]  time = 0.007, size = 43, normalized size = 0.8 \[{\frac{2}{3\,a}{x}^{{\frac{3}{2}}}}-2\,{\frac{b\sqrt{x}}{{a}^{2}}}+2\,{\frac{{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(a+b/x),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239363, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (a x - 3 \, b\right )} \sqrt{x}}{3 \, a^{2}}, \frac{2 \,{\left (3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) +{\left (a x - 3 \, b\right )} \sqrt{x}\right )}}{3 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.73743, size = 105, normalized size = 1.98 \[ \begin{cases} \frac{2 x^{\frac{3}{2}}}{3 a} - \frac{2 b \sqrt{x}}{a^{2}} - \frac{i b^{\frac{3}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{3} \sqrt{\frac{1}{a}}} + \frac{i b^{\frac{3}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{3} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{5}{2}}}{5 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(a+b/x),x)

[Out]

Piecewise((2*x**(3/2)/(3*a) - 2*b*sqrt(x)/a**2 - I*b**(3/2)*log(-I*sqrt(b)*sqrt(
1/a) + sqrt(x))/(a**3*sqrt(1/a)) + I*b**(3/2)*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))
/(a**3*sqrt(1/a)), Ne(a, 0)), (2*x**(5/2)/(5*b), True))

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GIAC/XCAS [A]  time = 0.223446, size = 61, normalized size = 1.15 \[ \frac{2 \, b^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{2 \,{\left (a^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + 2/3*(a^2*x^(3/2) - 3*a*b*sqr
t(x))/a^3