3.2413 \(\int (a+\frac{b}{\sqrt [3]{x}})^3 x^3 \, dx\)

Optimal. Leaf size=47 \[ \frac{9}{11} a^2 b x^{11/3}+\frac{a^3 x^4}{4}+\frac{9}{10} a b^2 x^{10/3}+\frac{b^3 x^3}{3} \]

[Out]

(b^3*x^3)/3 + (9*a*b^2*x^(10/3))/10 + (9*a^2*b*x^(11/3))/11 + (a^3*x^4)/4

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Rubi [A]  time = 0.0343602, 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} \[ \frac{9}{11} a^2 b x^{11/3}+\frac{a^3 x^4}{4}+\frac{9}{10} a b^2 x^{10/3}+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^3*x^3,x]

[Out]

(b^3*x^3)/3 + (9*a*b^2*x^(10/3))/10 + (9*a^2*b*x^(11/3))/11 + (a^3*x^4)/4

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 \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^3 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^3 x^2 \, dx\\ &=3 \operatorname{Subst}\left (\int x^8 (b+a x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^3 x^8+3 a b^2 x^9+3 a^2 b x^{10}+a^3 x^{11}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{b^3 x^3}{3}+\frac{9}{10} a b^2 x^{10/3}+\frac{9}{11} a^2 b x^{11/3}+\frac{a^3 x^4}{4}\\ \end{align*}

Mathematica [A]  time = 0.0234589, size = 41, normalized size = 0.87 \[ \frac{1}{660} x^3 \left (540 a^2 b x^{2/3}+165 a^3 x+594 a b^2 \sqrt [3]{x}+220 b^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^3*x^3,x]

[Out]

(x^3*(220*b^3 + 594*a*b^2*x^(1/3) + 540*a^2*b*x^(2/3) + 165*a^3*x))/660

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3}}+{\frac{9\,{b}^{2}a}{10}{x}^{{\frac{10}{3}}}}+{\frac{9\,b{a}^{2}}{11}{x}^{{\frac{11}{3}}}}+{\frac{{a}^{3}{x}^{4}}{4}} \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+9/10*a*b^2*x^(10/3)+9/11*a^2*b*x^(11/3)+1/4*a^3*x^4

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Maxima [A]  time = 0.981474, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{660} \,{\left (165 \, a^{3} + \frac{540 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{594 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{220 \, b^{3}}{x}\right )} x^{4} \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]

1/660*(165*a^3 + 540*a^2*b/x^(1/3) + 594*a*b^2/x^(2/3) + 220*b^3/x)*x^4

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Fricas [A]  time = 1.47566, size = 96, normalized size = 2.04 \begin{align*} \frac{1}{4} \, a^{3} x^{4} + \frac{9}{11} \, a^{2} b x^{\frac{11}{3}} + \frac{9}{10} \, a b^{2} x^{\frac{10}{3}} + \frac{1}{3} \, b^{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, algorithm="fricas")

[Out]

1/4*a^3*x^4 + 9/11*a^2*b*x^(11/3) + 9/10*a*b^2*x^(10/3) + 1/3*b^3*x^3

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Sympy [A]  time = 2.69634, size = 42, normalized size = 0.89 \begin{align*} \frac{a^{3} x^{4}}{4} + \frac{9 a^{2} b x^{\frac{11}{3}}}{11} + \frac{9 a b^{2} x^{\frac{10}{3}}}{10} + \frac{b^{3} x^{3}}{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*x**4/4 + 9*a**2*b*x**(11/3)/11 + 9*a*b**2*x**(10/3)/10 + b**3*x**3/3

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Giac [A]  time = 1.21503, size = 47, normalized size = 1. \begin{align*} \frac{1}{4} \, a^{3} x^{4} + \frac{9}{11} \, a^{2} b x^{\frac{11}{3}} + \frac{9}{10} \, a b^{2} x^{\frac{10}{3}} + \frac{1}{3} \, b^{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, algorithm="giac")

[Out]

1/4*a^3*x^4 + 9/11*a^2*b*x^(11/3) + 9/10*a*b^2*x^(10/3) + 1/3*b^3*x^3