\(\int \sqrt [3]{x} \log (x) \, dx\) [235]

Optimal result

Integrand size = 8, antiderivative size = 21 \[ \int \sqrt [3]{x} \log (x) \, dx=-\frac {9 x^{4/3}}{16}+\frac {3}{4} x^{4/3} \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int \sqrt [3]{x} \log (x) \, dx=\frac {3}{4} x^{4/3} \log (x)-\frac {9 x^{4/3}}{16} \]

[In]

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

[Out]

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

Rule 2341

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*} \text {integral}& = -\frac {9 x^{4/3}}{16}+\frac {3}{4} x^{4/3} \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{x} \log (x) \, dx=\frac {3}{16} x^{4/3} (-3+4 \log (x)) \]

[In]

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

[Out]

(3*x^(4/3)*(-3 + 4*Log[x]))/16

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(-\frac {9 x^{\frac {4}{3}}}{16}+\frac {3 x^{\frac {4}{3}} \ln \left (x \right )}{4}\)	\(14\)
default	\(-\frac {9 x^{\frac {4}{3}}}{16}+\frac {3 x^{\frac {4}{3}} \ln \left (x \right )}{4}\)	\(14\)
risch	\(-\frac {9 x^{\frac {4}{3}}}{16}+\frac {3 x^{\frac {4}{3}} \ln \left (x \right )}{4}\)	\(14\)

[In]

int(x^(1/3)*ln(x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \sqrt [3]{x} \log (x) \, dx=\frac {3}{16} \, {\left (4 \, x \log \left (x\right ) - 3 \, x\right )} x^{\frac {1}{3}} \]

[In]

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

[Out]

3/16*(4*x*log(x) - 3*x)*x^(1/3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (19) = 38\).

Time = 1.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.00 \[ \int \sqrt [3]{x} \log (x) \, dx=\begin {cases} - \frac {3 x^{\frac {4}{3}} \log {\left (\frac {1}{x} \right )}}{4} + \frac {3 x^{\frac {4}{3}} \log {\left (x \right )}}{4} - \frac {9 x^{\frac {4}{3}}}{8} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\frac {3 x^{\frac {4}{3}} \log {\left (x \right )}}{4} - \frac {9 x^{\frac {4}{3}}}{16} & \text {for}\: \left |{x}\right | < 1 \\- \frac {3 x^{\frac {4}{3}} \log {\left (\frac {1}{x} \right )}}{4} - \frac {9 x^{\frac {4}{3}}}{16} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {7}{3}, \frac {7}{3} \\\frac {4}{3}, \frac {4}{3} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {7}{3}, \frac {7}{3}, 1 & \\ & \frac {4}{3}, \frac {4}{3}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-3*x**(4/3)*log(1/x)/4 + 3*x**(4/3)*log(x)/4 - 9*x**(4/3)/8, (Abs(x) < 1) & (1/Abs(x) < 1)), (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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{x} \log (x) \, dx=\frac {3}{4} \, x^{\frac {4}{3}} \log \left (x\right ) - \frac {9}{16} \, x^{\frac {4}{3}} \]

[In]

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

[Out]

3/4*x^(4/3)*log(x) - 9/16*x^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt [3]{x} \log (x) \, dx=\frac {3}{4} \, x^{\frac {4}{3}} \log \left (x\right ) - \frac {9}{16} \, x^{\frac {4}{3}} \]

[In]

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

[Out]

3/4*x^(4/3)*log(x) - 9/16*x^(4/3)

Mupad [B] (verification not implemented)

Time = 1.48 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \sqrt [3]{x} \log (x) \, dx=\frac {3\,x^{4/3}\,\left (\ln \left (x\right )-\frac {3}{4}\right )}{4} \]

[In]

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

[Out]

(3*x^(4/3)*(log(x) - 3/4))/4