3.2397 \(\int (a+\frac{b}{\sqrt [3]{x}}) x \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^2}{2}+\frac{3}{5} b x^{5/3} \]

[Out]

(3*b*x^(5/3))/5 + (a*x^2)/2

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Rubi [A]  time = 0.0048784, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^2}{2}+\frac{3}{5} b x^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))*x,x]

[Out]

(3*b*x^(5/3))/5 + (a*x^2)/2

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

Mathematica [A]  time = 0.0026533, size = 19, normalized size = 1. \[ \frac{a x^2}{2}+\frac{3}{5} b x^{5/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))*x,x]

[Out]

(3*b*x^(5/3))/5 + (a*x^2)/2

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Maple [A]  time = 0., size = 14, normalized size = 0.7 \begin{align*}{\frac{3\,b}{5}{x}^{{\frac{5}{3}}}}+{\frac{a{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.970801, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{10} \,{\left (5 \, a + \frac{6 \, b}{x^{\frac{1}{3}}}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*a + 6*b/x^(1/3))*x^2

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Fricas [A]  time = 1.49118, size = 36, normalized size = 1.89 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{3}{5} \, b x^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + 3/5*b*x^(5/3)

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Sympy [A]  time = 0.206615, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{2}}{2} + \frac{3 b x^{\frac{5}{3}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + 3*b*x**(5/3)/5

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Giac [A]  time = 1.14602, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{3}{5} \, b x^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + 3/5*b*x^(5/3)