3.1735 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{2 a}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0572949, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.74981, size = 29, normalized size = 0.85 \[ - \frac{2 a}{b^{2} \sqrt{a + \frac{b}{x}}} - \frac{2 \sqrt{a + \frac{b}{x}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0340849, size = 29, normalized size = 0.85 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (2 a x+b)}{b^2 (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 31, normalized size = 0.9 \[ -2\,{\frac{ \left ( ax+b \right ) \left ( 2\,ax+b \right ) }{{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.43811, size = 41, normalized size = 1.21 \[ -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b^{2}} - \frac{2 \, a}{\sqrt{a + \frac{b}{x}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233898, size = 34, normalized size = 1. \[ -\frac{2 \,{\left (2 \, a x + b\right )}}{b^{2} x \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.73243, size = 42, normalized size = 1.24 \[ \begin{cases} - \frac{4 a}{b^{2} \sqrt{a + \frac{b}{x}}} - \frac{2}{b x \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{3}{2}} x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*a/(b**2*sqrt(a + b/x)) - 2/(b*x*sqrt(a + b/x)), Ne(b, 0)), (-1/(2*
a**(3/2)*x**2), True))

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GIAC/XCAS [A]  time = 0.256332, size = 47, normalized size = 1.38 \[ -2 \, b{\left (\frac{a}{b^{3} \sqrt{\frac{a x + b}{x}}} + \frac{\sqrt{\frac{a x + b}{x}}}{b^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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