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]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {a x^2}{2}+\frac {3}{5} b x^{5/3} \]

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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fricas [A]  time = 0.76, size = 13, normalized size = 0.68 \[ \frac {1}{2} \, a x^{2} + \frac {3}{5} \, b x^{\frac {5}{3}} \]

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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giac [A]  time = 0.19, size = 13, normalized size = 0.68 \[ \frac {1}{2} \, a x^{2} + \frac {3}{5} \, b x^{\frac {5}{3}} \]

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)

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ \frac {a \,x^{2}}{2}+\frac {3 b \,x^{\frac {5}{3}}}{5} \]

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 = 0.60, size = 15, normalized size = 0.79 \[ \frac {1}{10} \, {\left (5 \, a + \frac {6 \, b}{x^{\frac {1}{3}}}\right )} x^{2} \]

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \[ \frac {a\,x^2}{2}+\frac {3\,b\,x^{5/3}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b/x^(1/3)),x)

[Out]

(a*x^2)/2 + (3*b*x^(5/3))/5

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sympy [A]  time = 0.24, size = 15, normalized size = 0.79 \[ \frac {a x^{2}}{2} + \frac {3 b x^{\frac {5}{3}}}{5} \]

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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