3.1582 \(\int \frac{(a+\frac{b}{x})^3}{x^6} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]

[Out]

-b^3/(8*x^8) - (3*a*b^2)/(7*x^7) - (a^2*b)/(2*x^6) - a^3/(5*x^5)

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Rubi [A]  time = 0.0145501, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3/x^6,x]

[Out]

-b^3/(8*x^8) - (3*a*b^2)/(7*x^7) - (a^2*b)/(2*x^6) - a^3/(5*x^5)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^3}{x^6} \, dx &=\int \frac{(b+a x)^3}{x^9} \, dx\\ &=\int \left (\frac{b^3}{x^9}+\frac{3 a b^2}{x^8}+\frac{3 a^2 b}{x^7}+\frac{a^3}{x^6}\right ) \, dx\\ &=-\frac{b^3}{8 x^8}-\frac{3 a b^2}{7 x^7}-\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0059075, size = 43, normalized size = 1. \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^6,x]

[Out]

-b^3/(8*x^8) - (3*a*b^2)/(7*x^7) - (a^2*b)/(2*x^6) - a^3/(5*x^5)

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Maple [A]  time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{8\,{x}^{8}}}-{\frac{3\,{b}^{2}a}{7\,{x}^{7}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{{a}^{3}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^3/x^6,x)

[Out]

-1/8*b^3/x^8-3/7*a*b^2/x^7-1/2*a^2*b/x^6-1/5*a^3/x^5

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Maxima [A]  time = 0.985202, size = 47, normalized size = 1.09 \begin{align*} -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^6,x, algorithm="maxima")

[Out]

-1/280*(56*a^3*x^3 + 140*a^2*b*x^2 + 120*a*b^2*x + 35*b^3)/x^8

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Fricas [A]  time = 1.39167, size = 86, normalized size = 2. \begin{align*} -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^6,x, algorithm="fricas")

[Out]

-1/280*(56*a^3*x^3 + 140*a^2*b*x^2 + 120*a*b^2*x + 35*b^3)/x^8

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Sympy [A]  time = 0.413283, size = 37, normalized size = 0.86 \begin{align*} - \frac{56 a^{3} x^{3} + 140 a^{2} b x^{2} + 120 a b^{2} x + 35 b^{3}}{280 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**3/x**6,x)

[Out]

-(56*a**3*x**3 + 140*a**2*b*x**2 + 120*a*b**2*x + 35*b**3)/(280*x**8)

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Giac [A]  time = 1.20866, size = 47, normalized size = 1.09 \begin{align*} -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^6,x, algorithm="giac")

[Out]

-1/280*(56*a^3*x^3 + 140*a^2*b*x^2 + 120*a*b^2*x + 35*b^3)/x^8