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

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {2341} \begin {gather*} \frac {2}{3} x^{3/2} \log (x)-\frac {4 x^{3/2}}{9} \end {gather*}

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 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*} \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 \begin {gather*} \frac {2}{9} x^{3/2} (-2+3 \log (x)) \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.91, size = 98, normalized size = 4.67 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 x^{\frac {3}{2}} \left (-4-3 \text {Log}\left [\frac {1}{x}\right ]+3 \text {Log}\left [x\right ]\right )}{9},\text {Abs}\left [x\right ]<1\text {\&\&}\frac {1}{\text {Abs}\left [x\right ]}<1\right \},\left \{\frac {2 x^{\frac {3}{2}} \left (-2+3 \text {Log}\left [x\right ]\right )}{9},\text {Abs}\left [x\right ]<1\right \},\left \{\frac {2 x^{\frac {3}{2}} \left (-2-3 \text {Log}\left [\frac {1}{x}\right ]\right )}{9},\frac {1}{\text {Abs}\left [x\right ]}<1\right \}\right \},-\text {meijerg}\left [\left \{\left \{1\right \},\left \{\frac {5}{2},\frac {5}{2}\right \}\right \},\left \{\left \{\frac {3}{2},\frac {3}{2}\right \},\left \{0\right \}\right \},x\right ]+\text {meijerg}\left [\left \{\left \{\frac {5}{2},\frac {5}{2},1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{\frac {3}{2},\frac {3}{2},0\right \}\right \},x\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[x]*Log[x],x]')

[Out]

Piecewise[{{2 x ^ (3 / 2) (-4 - 3 Log[1 / x] + 3 Log[x]) / 9, Abs[x] < 1 && 1 / Abs[x] < 1}, {2 x ^ (3 / 2) (-
2 + 3 Log[x]) / 9, Abs[x] < 1}, {2 x ^ (3 / 2) (-2 - 3 Log[1 / x]) / 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]]

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Maple [A]
time = 0.04, size = 14, normalized size = 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	\(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \log \left (x\right ) - \frac {4}{9} \, x^{\frac {3}{2}} \end {gather*}

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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Fricas [A]
time = 0.32, size = 14, normalized size = 0.67 \begin {gather*} \frac {2}{9} \, {\left (3 \, x \log \left (x\right ) - 2 \, x\right )} \sqrt {x} \end {gather*}

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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Sympy [A]
time = 0.90, size = 105, normalized size = 5.00 \begin {gather*} \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} \end {gather*}

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(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))

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Giac [A]
time = 0.00, size = 27, normalized size = 1.29 \begin {gather*} \frac {2}{3} \sqrt {x} x \ln x-\frac {2\cdot 2 \sqrt {x} x}{3\cdot 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \frac {2\,x^{3/2}\,\left (\ln \left (x\right )-\frac {2}{3}\right )}{3} \end {gather*}

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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