3.2419 \(\int \frac{(a+\frac{b}{\sqrt [3]{x}})^3}{x^3} \, dx\)

Optimal. Leaf size=47 \[ -\frac{9 a^2 b}{7 x^{7/3}}-\frac{a^3}{2 x^2}-\frac{9 a b^2}{8 x^{8/3}}-\frac{b^3}{3 x^3} \]

[Out]

-b^3/(3*x^3) - (9*a*b^2)/(8*x^(8/3)) - (9*a^2*b)/(7*x^(7/3)) - a^3/(2*x^2)

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Rubi [A]  time = 0.0218488, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 43} \[ -\frac{9 a^2 b}{7 x^{7/3}}-\frac{a^3}{2 x^2}-\frac{9 a b^2}{8 x^{8/3}}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^3/x^3,x]

[Out]

-b^3/(3*x^3) - (9*a*b^2)/(8*x^(8/3)) - (9*a^2*b)/(7*x^(7/3)) - a^3/(2*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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/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}{\sqrt [3]{x}}\right )^3}{x^3} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^3}{x^4} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^3}{x^{10}}+\frac{3 a b^2}{x^9}+\frac{3 a^2 b}{x^8}+\frac{a^3}{x^7}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b^3}{3 x^3}-\frac{9 a b^2}{8 x^{8/3}}-\frac{9 a^2 b}{7 x^{7/3}}-\frac{a^3}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0163333, size = 41, normalized size = 0.87 \[ -\frac{216 a^2 b x^{2/3}+84 a^3 x+189 a b^2 \sqrt [3]{x}+56 b^3}{168 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^3/x^3,x]

[Out]

-(56*b^3 + 189*a*b^2*x^(1/3) + 216*a^2*b*x^(2/3) + 84*a^3*x)/(168*x^3)

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Maple [A]  time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{3\,{x}^{3}}}-{\frac{9\,{b}^{2}a}{8}{x}^{-{\frac{8}{3}}}}-{\frac{9\,b{a}^{2}}{7}{x}^{-{\frac{7}{3}}}}-{\frac{{a}^{3}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^3/x^3-9/8*a*b^2/x^(8/3)-9/7*a^2*b/x^(7/3)-1/2*a^3/x^2

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Maxima [B]  time = 0.973976, size = 132, normalized size = 2.81 \begin{align*} -\frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9}}{3 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a}{8 \, b^{6}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{2}}{7 \, b^{6}} + \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{3}}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{4}}{b^{6}} + \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{5}}{4 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a + b/x^(1/3))^9/b^6 + 15/8*(a + b/x^(1/3))^8*a/b^6 - 30/7*(a + b/x^(1/3))^7*a^2/b^6 + 5*(a + b/x^(1/3))
^6*a^3/b^6 - 3*(a + b/x^(1/3))^5*a^4/b^6 + 3/4*(a + b/x^(1/3))^4*a^5/b^6

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Fricas [A]  time = 1.48432, size = 97, normalized size = 2.06 \begin{align*} -\frac{84 \, a^{3} x + 216 \, a^{2} b x^{\frac{2}{3}} + 189 \, a b^{2} x^{\frac{1}{3}} + 56 \, b^{3}}{168 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(84*a^3*x + 216*a^2*b*x^(2/3) + 189*a*b^2*x^(1/3) + 56*b^3)/x^3

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Sympy [A]  time = 2.74207, size = 44, normalized size = 0.94 \begin{align*} - \frac{a^{3}}{2 x^{2}} - \frac{9 a^{2} b}{7 x^{\frac{7}{3}}} - \frac{9 a b^{2}}{8 x^{\frac{8}{3}}} - \frac{b^{3}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**3/(2*x**2) - 9*a**2*b/(7*x**(7/3)) - 9*a*b**2/(8*x**(8/3)) - b**3/(3*x**3)

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Giac [A]  time = 1.22113, size = 47, normalized size = 1. \begin{align*} -\frac{84 \, a^{3} x + 216 \, a^{2} b x^{\frac{2}{3}} + 189 \, a b^{2} x^{\frac{1}{3}} + 56 \, b^{3}}{168 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(84*a^3*x + 216*a^2*b*x^(2/3) + 189*a*b^2*x^(1/3) + 56*b^3)/x^3