3.2412 \(\int (a+\frac{b}{\sqrt [3]{x}})^3 x^4 \, dx\)

Optimal. Leaf size=47 \[ \frac{9}{14} a^2 b x^{14/3}+\frac{a^3 x^5}{5}+\frac{9}{13} a b^2 x^{13/3}+\frac{b^3 x^4}{4} \]

[Out]

(b^3*x^4)/4 + (9*a*b^2*x^(13/3))/13 + (9*a^2*b*x^(14/3))/14 + (a^3*x^5)/5

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Rubi [A]  time = 0.0371428, 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} \[ \frac{9}{14} a^2 b x^{14/3}+\frac{a^3 x^5}{5}+\frac{9}{13} a b^2 x^{13/3}+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^3*x^4,x]

[Out]

(b^3*x^4)/4 + (9*a*b^2*x^(13/3))/13 + (9*a^2*b*x^(14/3))/14 + (a^3*x^5)/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 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^4 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^3 x^3 \, dx\\ &=3 \operatorname{Subst}\left (\int x^{11} (b+a x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^3 x^{11}+3 a b^2 x^{12}+3 a^2 b x^{13}+a^3 x^{14}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{b^3 x^4}{4}+\frac{9}{13} a b^2 x^{13/3}+\frac{9}{14} a^2 b x^{14/3}+\frac{a^3 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0278343, size = 47, normalized size = 1. \[ \frac{9}{14} a^2 b x^{14/3}+\frac{a^3 x^5}{5}+\frac{9}{13} a b^2 x^{13/3}+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^3*x^4,x]

[Out]

(b^3*x^4)/4 + (9*a*b^2*x^(13/3))/13 + (9*a^2*b*x^(14/3))/14 + (a^3*x^5)/5

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{9\,{b}^{2}a}{13}{x}^{{\frac{13}{3}}}}+{\frac{9\,b{a}^{2}}{14}{x}^{{\frac{14}{3}}}}+{\frac{{a}^{3}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))^3*x^4,x)

[Out]

1/4*b^3*x^4+9/13*a*b^2*x^(13/3)+9/14*a^2*b*x^(14/3)+1/5*a^3*x^5

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Maxima [A]  time = 0.9572, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{1820} \,{\left (364 \, a^{3} + \frac{1170 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{1260 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{455 \, b^{3}}{x}\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^3*x^4,x, algorithm="maxima")

[Out]

1/1820*(364*a^3 + 1170*a^2*b/x^(1/3) + 1260*a*b^2/x^(2/3) + 455*b^3/x)*x^5

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Fricas [A]  time = 1.53258, size = 96, normalized size = 2.04 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{9}{14} \, a^{2} b x^{\frac{14}{3}} + \frac{9}{13} \, a b^{2} x^{\frac{13}{3}} + \frac{1}{4} \, b^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^3*x^4,x, algorithm="fricas")

[Out]

1/5*a^3*x^5 + 9/14*a^2*b*x^(14/3) + 9/13*a*b^2*x^(13/3) + 1/4*b^3*x^4

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Sympy [A]  time = 4.31274, size = 42, normalized size = 0.89 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{9 a^{2} b x^{\frac{14}{3}}}{14} + \frac{9 a b^{2} x^{\frac{13}{3}}}{13} + \frac{b^{3} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))**3*x**4,x)

[Out]

a**3*x**5/5 + 9*a**2*b*x**(14/3)/14 + 9*a*b**2*x**(13/3)/13 + b**3*x**4/4

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Giac [A]  time = 1.14833, size = 47, normalized size = 1. \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{9}{14} \, a^{2} b x^{\frac{14}{3}} + \frac{9}{13} \, a b^{2} x^{\frac{13}{3}} + \frac{1}{4} \, b^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^3*x^4,x, algorithm="giac")

[Out]

1/5*a^3*x^5 + 9/14*a^2*b*x^(14/3) + 9/13*a*b^2*x^(13/3) + 1/4*b^3*x^4