\(\int \sqrt {x} \log (x) \, dx\) [57]

Optimal result

Integrand size = 8, antiderivative size = 21 \[ \int \sqrt {x} \log (x) \, dx=-\frac {4 x^{3/2}}{9}+\frac {2}{3} x^{3/2} \log (x) \]

[Out]

-4/9*x^(3/2)+2/3*x^(3/2)*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} \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{3} x^{3/2} \log (x)-\frac {4 x^{3/2}}{9} \]

[In]

Int[Sqrt[x]*Log[x],x]

[Out]

(-4*x^(3/2))/9 + (2*x^(3/2)*Log[x])/3

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 {4 x^{3/2}}{9}+\frac {2}{3} x^{3/2} \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{9} x^{3/2} (-2+3 \log (x)) \]

[In]

Integrate[Sqrt[x]*Log[x],x]

[Out]

(2*x^(3/2)*(-2 + 3*Log[x]))/9

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\)	\(14\)
default	\(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\)	\(14\)
risch	\(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\)	\(14\)

[In]

int(ln(x)*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

-4/9*x^(3/2)+2/3*x^(3/2)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{9} \, {\left (3 \, x \log \left (x\right ) - 2 \, x\right )} \sqrt {x} \]

[In]

integrate(log(x)*x^(1/2),x, algorithm="fricas")

[Out]

2/9*(3*x*log(x) - 2*x)*sqrt(x)

Sympy [B] (verification not implemented)

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

Time = 0.93 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.00 \[ \int \sqrt {x} \log (x) \, dx=\begin {cases} - \frac {2 x^{\frac {3}{2}} \log {\left (\frac {1}{x} \right )}}{3} + \frac {2 x^{\frac {3}{2}} \log {\left (x \right )}}{3} - \frac {8 x^{\frac {3}{2}}}{9} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\frac {2 x^{\frac {3}{2}} \log {\left (x \right )}}{3} - \frac {4 x^{\frac {3}{2}}}{9} & \text {for}\: \left |{x}\right | < 1 \\- \frac {2 x^{\frac {3}{2}} \log {\left (\frac {1}{x} \right )}}{3} - \frac {4 x^{\frac {3}{2}}}{9} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {5}{2}, \frac {5}{2} \\\frac {3}{2}, \frac {3}{2} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {5}{2}, \frac {5}{2}, 1 & \\ & \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(ln(x)*x**(1/2),x)

[Out]

Piecewise((-2*x**(3/2)*log(1/x)/3 + 2*x**(3/2)*log(x)/3 - 8*x**(3/2)/9, (Abs(x) < 1) & (1/Abs(x) < 1)), (2*x**
(3/2)*log(x)/3 - 4*x**(3/2)/9, Abs(x) < 1), (-2*x**(3/2)*log(1/x)/3 - 4*x**(3/2)/9, 1/Abs(x) < 1), (-meijerg((
(1,), (5/2, 5/2)), ((3/2, 3/2), (0,)), x) + meijerg(((5/2, 5/2, 1), ()), ((), (3/2, 3/2, 0)), x), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \log \left (x\right ) - \frac {4}{9} \, x^{\frac {3}{2}} \]

[In]

integrate(log(x)*x^(1/2),x, algorithm="maxima")

[Out]

2/3*x^(3/2)*log(x) - 4/9*x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \log \left (x\right ) - \frac {4}{9} \, x^{\frac {3}{2}} \]

[In]

integrate(log(x)*x^(1/2),x, algorithm="giac")

[Out]

2/3*x^(3/2)*log(x) - 4/9*x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \sqrt {x} \log (x) \, dx=\frac {2\,x^{3/2}\,\left (\ln \left (x\right )-\frac {2}{3}\right )}{3} \]

[In]

int(x^(1/2)*log(x),x)

[Out]

(2*x^(3/2)*(log(x) - 2/3))/3