3.57 \(\int \sqrt {x} \log (x) \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{3} x^{3/2} \log (x)-\frac {4 x^{3/2}}{9} \]

[Out]

-4/9*x^(3/2)+2/3*x^(3/2)*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} \[ \frac {2}{3} x^{3/2} \log (x)-\frac {4 x^{3/2}}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*x^(3/2))/9 + (2*x^(3/2)*Log[x])/3

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 {x} \log (x) \, dx &=-\frac {4 x^{3/2}}{9}+\frac {2}{3} x^{3/2} \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.71 \[ \frac {2}{9} x^{3/2} (3 \log (x)-2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(-2 + 3*Log[x]))/9

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fricas [A]  time = 0.42, size = 14, normalized size = 0.67 \[ \frac {2}{9} \, {\left (3 \, x \log \relax (x) - 2 \, x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*x*log(x) - 2*x)*sqrt(x)

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giac [A]  time = 1.09, size = 13, normalized size = 0.62 \[ \frac {2}{3} \, x^{\frac {3}{2}} \log \relax (x) - \frac {4}{9} \, x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*log(x) - 4/9*x^(3/2)

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maple [A]  time = 0.00, size = 14, normalized size = 0.67 \[ \frac {2 x^{\frac {3}{2}} \ln \relax (x )}{3}-\frac {4 x^{\frac {3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 13, normalized size = 0.62 \[ \frac {2}{3} \, x^{\frac {3}{2}} \log \relax (x) - \frac {4}{9} \, x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*log(x) - 4/9*x^(3/2)

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mupad [B]  time = 0.02, size = 9, normalized size = 0.43 \[ \frac {2\,x^{3/2}\,\left (\ln \relax (x)-\frac {2}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^(3/2)*(log(x) - 2/3))/3

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sympy [A]  time = 1.94, size = 66, normalized size = 3.14 \[ \begin {cases} \frac {2 x^{\frac {3}{2}} \log {\relax (x )}}{3} - \frac {4 x^{\frac {3}{2}}}{9} & \text {for}\: \left |{x}\right | < 1 \\- \frac {2 x^{\frac {3}{2}} \log {\left (\frac {1}{x} \right )}}{3} - \frac {4 x^{\frac {3}{2}}}{9} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {5}{2}, \frac {5}{2} \\\frac {3}{2}, \frac {3}{2} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {5}{2}, \frac {5}{2}, 1 & \\ & \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((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))

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