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]

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

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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 = 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*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \[ \frac {a x^3}{3}+\frac {3}{8} b x^{8/3} \]

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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fricas [A]  time = 0.84, size = 13, normalized size = 0.68 \[ \frac {1}{3} \, a x^{3} + \frac {3}{8} \, b x^{\frac {8}{3}} \]

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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giac [A]  time = 0.15, size = 13, normalized size = 0.68 \[ \frac {1}{3} \, a x^{3} + \frac {3}{8} \, b x^{\frac {8}{3}} \]

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)

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ \frac {a \,x^{3}}{3}+\frac {3 b \,x^{\frac {8}{3}}}{8} \]

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 = 0.50, size = 15, normalized size = 0.79 \[ \frac {1}{24} \, {\left (8 \, a + \frac {9 \, b}{x^{\frac {1}{3}}}\right )} x^{3} \]

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \[ \frac {a\,x^3}{3}+\frac {3\,b\,x^{8/3}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b/x^(1/3)),x)

[Out]

(a*x^3)/3 + (3*b*x^(8/3))/8

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sympy [A]  time = 0.55, size = 15, normalized size = 0.79 \[ \frac {a x^{3}}{3} + \frac {3 b x^{\frac {8}{3}}}{8} \]

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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