3.1746 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

2/(3*b*(a + b/x)^(3/2))

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Rubi [A]  time = 0.0056434, antiderivative size = 18, 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} \[ \frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^2),x]

[Out]

2/(3*b*(a + b/x)^(3/2))

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 )^{5/2} x^2} \, dx &=\frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0091008, size = 18, normalized size = 1. \[ \frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^2),x]

[Out]

2/(3*b*(a + b/x)^(3/2))

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Maple [A]  time = 0.004, size = 25, normalized size = 1.4 \begin{align*}{\frac{2\,ax+2\,b}{3\,bx} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^2,x)

[Out]

2/3*(a*x+b)/x/b/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 1.00209, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^2,x, algorithm="maxima")

[Out]

2/3/((a + b/x)^(3/2)*b)

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Fricas [B]  time = 1.4152, size = 77, normalized size = 4.28 \begin{align*} \frac{2 \, x^{2} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.29215, size = 39, normalized size = 2.17 \begin{align*} \begin{cases} \frac{2}{3 a b \sqrt{a + \frac{b}{x}} + \frac{3 b^{2} \sqrt{a + \frac{b}{x}}}{x}} & \text{for}\: b \neq 0 \\- \frac{1}{a^{\frac{5}{2}} x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**2,x)

[Out]

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

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Giac [A]  time = 1.28378, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^2,x, algorithm="giac")

[Out]

2/3/((a + b/x)^(3/2)*b)