3.1787 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2 \sqrt{x}}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.0531805, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 b}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2 \sqrt{x}}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*Sqrt[x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.29511, size = 36, normalized size = 0.82 \[ \frac{2 \sqrt{x}}{a \sqrt{a + \frac{b}{x}}} + \frac{4 b}{a^{2} \sqrt{x} \sqrt{a + \frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(3/2)/x**(1/2),x)

[Out]

2*sqrt(x)/(a*sqrt(a + b/x)) + 4*b/(a**2*sqrt(x)*sqrt(a + b/x))

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Mathematica [A]  time = 0.0438773, size = 35, normalized size = 0.8 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+2 b)}{a^2 (a x+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(3/2)*Sqrt[x]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 0.7 \[ 2\,{\frac{ \left ( ax+b \right ) \left ( ax+2\,b \right ) }{{a}^{2}{x}^{3/2}} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^(1/2),x)

[Out]

2*(a*x+b)*(a*x+2*b)/a^2/x^(3/2)/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 1.44857, size = 49, normalized size = 1.11 \[ \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a^{2}} + \frac{2 \, b}{\sqrt{a + \frac{b}{x}} a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*sqrt(x)),x, algorithm="maxima")

[Out]

2*sqrt(a + b/x)*sqrt(x)/a^2 + 2*b/(sqrt(a + b/x)*a^2*sqrt(x))

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Fricas [A]  time = 0.234281, size = 35, normalized size = 0.8 \[ \frac{2 \,{\left (a x + 2 \, b\right )}}{a^{2} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*sqrt(x)),x, algorithm="fricas")

[Out]

2*(a*x + 2*b)/(a^2*sqrt(x)*sqrt((a*x + b)/x))

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Sympy [A]  time = 26.7802, size = 39, normalized size = 0.89 \[ \frac{2 x}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(3/2)/x**(1/2),x)

[Out]

2*x/(a*sqrt(b)*sqrt(a*x/b + 1)) + 4*sqrt(b)/(a**2*sqrt(a*x/b + 1))

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GIAC/XCAS [A]  time = 0.229112, size = 42, normalized size = 0.95 \[ \frac{2 \,{\left (\sqrt{a x + b} + \frac{b}{\sqrt{a x + b}}\right )}}{a^{2}} - \frac{4 \, \sqrt{b}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*sqrt(x)),x, algorithm="giac")

[Out]

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