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

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

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Rubi [A]  time = 0.00, antiderivative size = 13, 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} \[ -\frac {1}{\sqrt [3]{1-\frac {3}{x}}} \]

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*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ -\frac {1}{\sqrt [3]{\frac {x-3}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 3/x)^(4/3)*x^2),x]

[Out]

-((-3 + x)/x)^(-1/3)

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IntegrateAlgebraic [A]  time = 0.01, size = 19, normalized size = 1.46 \[ -\frac {\left (\frac {x-3}{x}\right )^{2/3} x}{x-3} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 - 3/x)^(4/3)*x^2),x]

[Out]

-((((-3 + x)/x)^(2/3)*x)/(-3 + x))

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fricas [A]  time = 0.52, size = 17, normalized size = 1.31 \[ -\frac {x \left (\frac {x - 3}{x}\right )^{\frac {2}{3}}}{x - 3} \]

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)

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giac [A]  time = 0.65, size = 11, normalized size = 0.85 \[ -\frac {1}{\left (\frac {x - 3}{x}\right )^{\frac {1}{3}}} \]

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/((x - 3)/x)^(1/3)

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maple [A]  time = 0.03, size = 12, normalized size = 0.92

method	result	size
derivativedivides	\(-\frac {1}{\left (1-\frac {3}{x}\right )^{\frac {1}{3}}}\)	\(12\)
default	\(-\frac {1}{\left (1-\frac {3}{x}\right )^{\frac {1}{3}}}\)	\(12\)
risch	\(-\frac {1}{\left (\frac {-3+x}{x}\right )^{\frac {1}{3}}}\)	\(12\)
gosper	\(-\frac {-3+x}{x \left (\frac {-3+x}{x}\right )^{\frac {4}{3}}}\)	\(18\)
trager	\(-\frac {x \left (-\frac {3-x}{x}\right )^{\frac {2}{3}}}{-3+x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(1-3/x)^(1/3)

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maxima [A]  time = 0.52, size = 11, normalized size = 0.85 \[ -\frac {1}{{\left (-\frac {3}{x} + 1\right )}^{\frac {1}{3}}} \]

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)

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mupad [B]  time = 0.46, size = 11, normalized size = 0.85 \[ -\frac {1}{{\left (1-\frac {3}{x}\right )}^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(1 - 3/x)^(4/3)),x)

[Out]

-1/(1 - 3/x)^(1/3)

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sympy [A]  time = 0.98, size = 10, normalized size = 0.77 \[ - \frac {1}{\sqrt [3]{1 - \frac {3}{x}}} \]

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)

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