3.293 \(\int \frac{1}{(1-\frac{3}{x})^{4/3} x^2} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{\sqrt [3]{1-\frac{3}{x}}} \]

[Out]

-(1 - 3/x)^(-1/3)

________________________________________________________________________________________

Rubi [A]  time = 0.0037591, antiderivative size = 13, 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} \[ -\frac{1}{\sqrt [3]{1-\frac{3}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 3/x)^(4/3)*x^2),x]

[Out]

-(1 - 3/x)^(-1/3)

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 (1-\frac{3}{x}\right )^{4/3} x^2} \, dx &=-\frac{1}{\sqrt [3]{1-\frac{3}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0054422, size = 13, normalized size = 1. \[ -\frac{1}{\sqrt [3]{\frac{x-3}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 3/x)^(4/3)*x^2),x]

[Out]

-((-3 + x)/x)^(-1/3)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 18, normalized size = 1.4 \begin{align*} -{\frac{-3+x}{x} \left ({\frac{-3+x}{x}} \right ) ^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-3/x)^(4/3)/x^2,x)

[Out]

-(-3+x)/x/((-3+x)/x)^(4/3)

________________________________________________________________________________________

Maxima [A]  time = 0.953112, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{{\left (-\frac{3}{x} + 1\right )}^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-3/x)^(4/3)/x^2,x, algorithm="maxima")

[Out]

-1/(-3/x + 1)^(1/3)

________________________________________________________________________________________

Fricas [A]  time = 1.78042, size = 41, normalized size = 3.15 \begin{align*} -\frac{x \left (\frac{x - 3}{x}\right )^{\frac{2}{3}}}{x - 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-3/x)^(4/3)/x^2,x, algorithm="fricas")

[Out]

-x*((x - 3)/x)^(2/3)/(x - 3)

________________________________________________________________________________________

Sympy [A]  time = 0.916459, size = 10, normalized size = 0.77 \begin{align*} - \frac{1}{\sqrt [3]{1 - \frac{3}{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-3/x)**(4/3)/x**2,x)

[Out]

-1/(1 - 3/x)**(1/3)

________________________________________________________________________________________

Giac [A]  time = 1.06431, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{{\left (-\frac{3}{x} + 1\right )}^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-3/x)^(4/3)/x^2,x, algorithm="giac")

[Out]

-1/(-3/x + 1)^(1/3)