3.235 \(\int \sqrt [3]{x} \log (x) \, dx\)

Optimal. Leaf size=21 \[ \frac {3}{4} x^{4/3} \log (x)-\frac {9 x^{4/3}}{16} \]

[Out]

-9/16*x^(4/3)+3/4*x^(4/3)*ln(x)

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2304} \[ \frac {3}{4} x^{4/3} \log (x)-\frac {9 x^{4/3}}{16} \]

Antiderivative was successfully verified.

[In]

Int[x^(1/3)*Log[x],x]

[Out]

(-9*x^(4/3))/16 + (3*x^(4/3)*Log[x])/4

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {align*} \int \sqrt [3]{x} \log (x) \, dx &=-\frac {9 x^{4/3}}{16}+\frac {3}{4} x^{4/3} \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.71 \[ \frac {3}{16} x^{4/3} (4 \log (x)-3) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1/3)*Log[x],x]

[Out]

(3*x^(4/3)*(-3 + 4*Log[x]))/16

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fricas [A]  time = 0.46, size = 14, normalized size = 0.67 \[ \frac {3}{16} \, {\left (4 \, x \log \relax (x) - 3 \, x\right )} x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/3)*log(x),x, algorithm="fricas")

[Out]

3/16*(4*x*log(x) - 3*x)*x^(1/3)

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giac [A]  time = 0.16, size = 13, normalized size = 0.62 \[ \frac {3}{4} \, x^{\frac {4}{3}} \log \relax (x) - \frac {9}{16} \, x^{\frac {4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/3)*log(x),x, algorithm="giac")

[Out]

3/4*x^(4/3)*log(x) - 9/16*x^(4/3)

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maple [A]  time = 0.07, size = 14, normalized size = 0.67 \[ \frac {3 x^{\frac {4}{3}} \ln \relax (x )}{4}-\frac {9 x^{\frac {4}{3}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/3)*ln(x),x)

[Out]

-9/16*x^(4/3)+3/4*x^(4/3)*ln(x)

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maxima [A]  time = 0.44, size = 13, normalized size = 0.62 \[ \frac {3}{4} \, x^{\frac {4}{3}} \log \relax (x) - \frac {9}{16} \, x^{\frac {4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/3)*log(x),x, algorithm="maxima")

[Out]

3/4*x^(4/3)*log(x) - 9/16*x^(4/3)

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mupad [B]  time = 0.34, size = 9, normalized size = 0.43 \[ \frac {3\,x^{4/3}\,\left (\ln \relax (x)-\frac {3}{4}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/3)*log(x),x)

[Out]

(3*x^(4/3)*(log(x) - 3/4))/4

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sympy [A]  time = 2.61, size = 66, normalized size = 3.14 \[ \begin {cases} \frac {3 x^{\frac {4}{3}} \log {\relax (x )}}{4} - \frac {9 x^{\frac {4}{3}}}{16} & \text {for}\: \left |{x}\right | < 1 \\- \frac {3 x^{\frac {4}{3}} \log {\left (\frac {1}{x} \right )}}{4} - \frac {9 x^{\frac {4}{3}}}{16} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {7}{3}, \frac {7}{3} \\\frac {4}{3}, \frac {4}{3} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {7}{3}, \frac {7}{3}, 1 & \\ & \frac {4}{3}, \frac {4}{3}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/3)*ln(x),x)

[Out]

Piecewise((3*x**(4/3)*log(x)/4 - 9*x**(4/3)/16, Abs(x) < 1), (-3*x**(4/3)*log(1/x)/4 - 9*x**(4/3)/16, 1/Abs(x)
< 1), (-meijerg(((1,), (7/3, 7/3)), ((4/3, 4/3), (0,)), x) + meijerg(((7/3, 7/3, 1), ()), ((), (4/3, 4/3, 0))
, x), True))

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