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

________________________________________________________________________________________

Rubi [A]  time = 0.0067835, antiderivative size = 21, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0023618, 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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \begin{align*} -{\frac{9}{16}{x}^{{\frac{4}{3}}}}+{\frac{3\,\ln \left ( x \right ) }{4}{x}^{{\frac{4}{3}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.06314, size = 18, normalized size = 0.86 \begin{align*} \frac{3}{4} \, x^{\frac{4}{3}} \log \left (x\right ) - \frac{9}{16} \, x^{\frac{4}{3}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.93388, size = 45, normalized size = 2.14 \begin{align*} \frac{3}{16} \,{\left (4 \, x \log \left (x\right ) - 3 \, x\right )} x^{\frac{1}{3}} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 3.09549, size = 66, normalized size = 3.14 \begin{align*} \begin{cases} \frac{3 x^{\frac{4}{3}} \log{\left (x \right )}}{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} \end{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.31856, size = 18, normalized size = 0.86 \begin{align*} \frac{3}{4} \, x^{\frac{4}{3}} \log \left (x\right ) - \frac{9}{16} \, x^{\frac{4}{3}} \end{align*}

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)