3.1640 \(\int \frac{1}{(a+\frac{b}{x})^3 x^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{2 a (a x+b)^2} \]

[Out]

-1/(2*a*(b + a*x)^2)

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Rubi [A]  time = 0.0037481, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 32} \[ -\frac{1}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^3),x]

[Out]

-1/(2*a*(b + a*x)^2)

Rule 263

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0024803, size = 14, normalized size = 1. \[ -\frac{1}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^3),x]

[Out]

-1/(2*a*(b + a*x)^2)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ( ax+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961641, size = 32, normalized size = 2.29 \begin{align*} -\frac{1}{2 \,{\left (a^{3} x^{2} + 2 \, a^{2} b x + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a^3*x^2 + 2*a^2*b*x + a*b^2)

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Fricas [A]  time = 1.64829, size = 49, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (a^{3} x^{2} + 2 \, a^{2} b x + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a^3*x^2 + 2*a^2*b*x + a*b^2)

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Sympy [B]  time = 0.316896, size = 26, normalized size = 1.86 \begin{align*} - \frac{1}{2 a^{3} x^{2} + 4 a^{2} b x + 2 a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a**3*x**2 + 4*a**2*b*x + 2*a*b**2)

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Giac [A]  time = 1.13612, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (a x + b\right )}^{2} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a*x + b)^2*a)