3.1580 \(\int \frac{(a+\frac{b}{x})^3}{x^4} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{6 x^6} \]

[Out]

-b^3/(6*x^6) - (3*a*b^2)/(5*x^5) - (3*a^2*b)/(4*x^4) - a^3/(3*x^3)

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Rubi [A]  time = 0.0149324, 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} \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3/x^4,x]

[Out]

-b^3/(6*x^6) - (3*a*b^2)/(5*x^5) - (3*a^2*b)/(4*x^4) - a^3/(3*x^3)

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^4} \, dx &=\int \frac{(b+a x)^3}{x^7} \, dx\\ &=\int \left (\frac{b^3}{x^7}+\frac{3 a b^2}{x^6}+\frac{3 a^2 b}{x^5}+\frac{a^3}{x^4}\right ) \, dx\\ &=-\frac{b^3}{6 x^6}-\frac{3 a b^2}{5 x^5}-\frac{3 a^2 b}{4 x^4}-\frac{a^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0034169, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3/x^4,x]

[Out]

-b^3/(6*x^6) - (3*a*b^2)/(5*x^5) - (3*a^2*b)/(4*x^4) - a^3/(3*x^3)

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{6\,{x}^{6}}}-{\frac{3\,{b}^{2}a}{5\,{x}^{5}}}-{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-{\frac{{a}^{3}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^3/x^4,x)

[Out]

-1/6*b^3/x^6-3/5*a*b^2/x^5-3/4*a^2*b/x^4-1/3*a^3/x^3

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Maxima [A]  time = 0.969338, size = 47, normalized size = 1.09 \begin{align*} -\frac{20 \, a^{3} x^{3} + 45 \, a^{2} b x^{2} + 36 \, a b^{2} x + 10 \, b^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^4,x, algorithm="maxima")

[Out]

-1/60*(20*a^3*x^3 + 45*a^2*b*x^2 + 36*a*b^2*x + 10*b^3)/x^6

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Fricas [A]  time = 1.38942, size = 82, normalized size = 1.91 \begin{align*} -\frac{20 \, a^{3} x^{3} + 45 \, a^{2} b x^{2} + 36 \, a b^{2} x + 10 \, b^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^4,x, algorithm="fricas")

[Out]

-1/60*(20*a^3*x^3 + 45*a^2*b*x^2 + 36*a*b^2*x + 10*b^3)/x^6

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Sympy [A]  time = 0.382102, size = 37, normalized size = 0.86 \begin{align*} - \frac{20 a^{3} x^{3} + 45 a^{2} b x^{2} + 36 a b^{2} x + 10 b^{3}}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**3/x**4,x)

[Out]

-(20*a**3*x**3 + 45*a**2*b*x**2 + 36*a*b**2*x + 10*b**3)/(60*x**6)

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Giac [A]  time = 1.12491, size = 47, normalized size = 1.09 \begin{align*} -\frac{20 \, a^{3} x^{3} + 45 \, a^{2} b x^{2} + 36 \, a b^{2} x + 10 \, b^{3}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3/x^4,x, algorithm="giac")

[Out]

-1/60*(20*a^3*x^3 + 45*a^2*b*x^2 + 36*a*b^2*x + 10*b^3)/x^6