3.2416 \(\int (a+\frac{b}{\sqrt [3]{x}})^3 \, dx\)

Optimal. Leaf size=36 \[ \frac{9}{2} a^2 b x^{2/3}+a^3 x+9 a b^2 \sqrt [3]{x}+b^3 \log (x) \]

[Out]

9*a*b^2*x^(1/3) + (9*a^2*b*x^(2/3))/2 + a^3*x + b^3*Log[x]

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Rubi [A]  time = 0.019956, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{9}{2} a^2 b x^{2/3}+a^3 x+9 a b^2 \sqrt [3]{x}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^3,x]

[Out]

9*a*b^2*x^(1/3) + (9*a^2*b*x^(2/3))/2 + a^3*x + b^3*Log[x]

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 \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^4} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}+\frac{3 a^2 b}{x^3}+\frac{3 a b^2}{x^2}+\frac{b^3}{x}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=9 a b^2 \sqrt [3]{x}+\frac{9}{2} a^2 b x^{2/3}+a^3 x+b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0146624, size = 36, normalized size = 1. \[ \frac{9}{2} a^2 b x^{2/3}+a^3 x+9 a b^2 \sqrt [3]{x}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^3,x]

[Out]

9*a*b^2*x^(1/3) + (9*a^2*b*x^(2/3))/2 + a^3*x + b^3*Log[x]

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Maple [A]  time = 0.001, size = 31, normalized size = 0.9 \begin{align*} 9\,a{b}^{2}\sqrt [3]{x}+{\frac{9\,b{a}^{2}}{2}{x}^{{\frac{2}{3}}}}+{a}^{3}x+{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0091, size = 41, normalized size = 1.14 \begin{align*} a^{3} x + b^{3} \log \left (x\right ) + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50694, size = 89, normalized size = 2.47 \begin{align*} a^{3} x + 3 \, b^{3} \log \left (x^{\frac{1}{3}}\right ) + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.290192, size = 36, normalized size = 1. \begin{align*} a^{3} x + \frac{9 a^{2} b x^{\frac{2}{3}}}{2} + 9 a b^{2} \sqrt [3]{x} + b^{3} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + 9*a**2*b*x**(2/3)/2 + 9*a*b**2*x**(1/3) + b**3*log(x)

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Giac [A]  time = 1.16701, size = 42, normalized size = 1.17 \begin{align*} a^{3} x + b^{3} \log \left ({\left | x \right |}\right ) + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x + b^3*log(abs(x)) + 9/2*a^2*b*x^(2/3) + 9*a*b^2*x^(1/3)