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/(a+b/x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \frac {2}{b \sqrt {a+\frac {b}{x}}} \]

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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fricas [A]  time = 1.01, size = 24, normalized size = 1.50 \[ \frac {2 \, x \sqrt {\frac {a x + b}{x}}}{a b x + b^{2}} \]

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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giac [A]  time = 0.16, size = 16, normalized size = 1.00 \[ \frac {2}{b \sqrt {\frac {a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(b*sqrt((a*x + b)/x))

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maple [A]  time = 0.00, size = 25, normalized size = 1.56 \[ \frac {2 a x +2 b}{\left (\frac {a x +b}{x}\right )^{\frac {3}{2}} b x} \]

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 = 0.99, size = 14, normalized size = 0.88 \[ \frac {2}{\sqrt {a + \frac {b}{x}} b} \]

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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mupad [B]  time = 1.31, size = 14, normalized size = 0.88 \[ \frac {2}{b\,\sqrt {a+\frac {b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b/x)^(3/2)),x)

[Out]

2/(b*(a + b/x)^(1/2))

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sympy [A]  time = 1.80, size = 20, normalized size = 1.25 \[ \begin {cases} \frac {2}{b \sqrt {a + \frac {b}{x}}} & \text {for}\: b \neq 0 \\- \frac {1}{a^{\frac {3}{2}} x} & \text {otherwise} \end {cases} \]

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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