3.2396 \(\int (a+\frac{b}{\sqrt [3]{x}}) x^2 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^3}{3}+\frac{3}{8} b x^{8/3} \]

[Out]

(3*b*x^(8/3))/8 + (a*x^3)/3

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Rubi [A]  time = 0.0049467, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{a x^3}{3}+\frac{3}{8} b x^{8/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))*x^2,x]

[Out]

(3*b*x^(8/3))/8 + (a*x^3)/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{\sqrt [3]{x}}\right ) x^2 \, dx &=\int \left (b x^{5/3}+a x^2\right ) \, dx\\ &=\frac{3}{8} b x^{8/3}+\frac{a x^3}{3}\\ \end{align*}

Mathematica [A]  time = 0.0028563, size = 19, normalized size = 1. \[ \frac{a x^3}{3}+\frac{3}{8} b x^{8/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))*x^2,x]

[Out]

(3*b*x^(8/3))/8 + (a*x^3)/3

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{3\,b}{8}{x}^{{\frac{8}{3}}}}+{\frac{a{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))*x^2,x)

[Out]

3/8*b*x^(8/3)+1/3*a*x^3

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Maxima [A]  time = 0.962582, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{24} \,{\left (8 \, a + \frac{9 \, b}{x^{\frac{1}{3}}}\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))*x^2,x, algorithm="maxima")

[Out]

1/24*(8*a + 9*b/x^(1/3))*x^3

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Fricas [A]  time = 1.70329, size = 36, normalized size = 1.89 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{3}{8} \, b x^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))*x^2,x, algorithm="fricas")

[Out]

1/3*a*x^3 + 3/8*b*x^(8/3)

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Sympy [A]  time = 0.472396, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{3}}{3} + \frac{3 b x^{\frac{8}{3}}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))*x**2,x)

[Out]

a*x**3/3 + 3*b*x**(8/3)/8

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Giac [A]  time = 1.20384, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{3}{8} \, b x^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))*x^2,x, algorithm="giac")

[Out]

1/3*a*x^3 + 3/8*b*x^(8/3)