3.2415 \(\int (a+\frac{b}{\sqrt [3]{x}})^3 x \, dx\)

Optimal. Leaf size=42 \[ \frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]

[Out]

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

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Rubi [A]  time = 0.0235856, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 190, 43} \[ \frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^3*x,x]

[Out]

b^3*x + (9*a*b^2*x^(4/3))/4 + (9*a^2*b*x^(5/3))/5 + (a^3*x^2)/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 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[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 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^3 \, dx\\ &=3 \operatorname{Subst}\left (\int x^2 (b+a x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (b^3 x^2+3 a b^2 x^3+3 a^2 b x^4+a^3 x^5\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=b^3 x+\frac{9}{4} a b^2 x^{4/3}+\frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}\\ \end{align*}

Mathematica [A]  time = 0.0158789, size = 42, normalized size = 1. \[ \frac{9}{5} a^2 b x^{5/3}+\frac{a^3 x^2}{2}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^3*x,x]

[Out]

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

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Maple [A]  time = 0.001, size = 33, normalized size = 0.8 \begin{align*}{b}^{3}x+{\frac{9\,{b}^{2}a}{4}{x}^{{\frac{4}{3}}}}+{\frac{9\,b{a}^{2}}{5}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}{a}^{3}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966784, size = 50, normalized size = 1.19 \begin{align*} \frac{1}{20} \,{\left (10 \, a^{3} + \frac{36 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{45 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{20 \, b^{3}}{x}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.41798, size = 82, normalized size = 1.95 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.644069, size = 39, normalized size = 0.93 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{5}{3}}}{5} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19377, size = 43, normalized size = 1.02 \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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