3.1736 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]

[Out]

2/(b*Sqrt[a + b/x])

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Rubi [A]  time = 0.0056079, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^2),x]

[Out]

2/(b*Sqrt[a + b/x])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^2} \, dx &=\frac{2}{b \sqrt{a+\frac{b}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0085634, size = 16, normalized size = 1. \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*x^2),x]

[Out]

2/(b*Sqrt[a + b/x])

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Maple [A]  time = 0.003, size = 25, normalized size = 1.6 \begin{align*} 2\,{\frac{ax+b}{bx} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.980123, size = 19, normalized size = 1.19 \begin{align*} \frac{2}{\sqrt{a + \frac{b}{x}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sqrt(a + b/x)*b)

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Fricas [A]  time = 1.46794, size = 50, normalized size = 3.12 \begin{align*} \frac{2 \, x \sqrt{\frac{a x + b}{x}}}{a b x + b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*sqrt((a*x + b)/x)/(a*b*x + b^2)

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Sympy [A]  time = 1.69579, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} \frac{2}{b \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{a^{\frac{3}{2}} x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.24935, size = 19, normalized size = 1.19 \begin{align*} \frac{2}{\sqrt{a + \frac{b}{x}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sqrt(a + b/x)*b)