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*x^(3/2))/9 + (2*x^(3/2)*Log[x])/3

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Rubi [A]  time = 0.0114925, 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 \[ \frac{2}{3} x^{3/2} \log (x)-\frac{4 x^{3/2}}{9} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Log[x],x]

[Out]

(-4*x^(3/2))/9 + (2*x^(3/2)*Log[x])/3

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Rubi in Sympy [A]  time = 1.07153, size = 19, normalized size = 0.9 \[ \frac{2 x^{\frac{3}{2}} \log{\left (x \right )}}{3} - \frac{4 x^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x)*x**(1/2),x)

[Out]

2*x**(3/2)*log(x)/3 - 4*x**(3/2)/9

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Mathematica [A]  time = 0.0041041, 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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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \[ -{\frac{4}{9}{x}^{{\frac{3}{2}}}}+{\frac{2\,\ln \left ( x \right ) }{3}{x}^{{\frac{3}{2}}}} \]

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 = 1.54764, size = 18, normalized size = 0.86 \[ \frac{2}{3} \, x^{\frac{3}{2}} \log \left (x\right ) - \frac{4}{9} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)*log(x),x, algorithm="maxima")

[Out]

2/3*x^(3/2)*log(x) - 4/9*x^(3/2)

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Fricas [A]  time = 0.211946, size = 19, normalized size = 0.9 \[ \frac{2}{9} \,{\left (3 \, x \log \left (x\right ) - 2 \, x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)*log(x),x, algorithm="fricas")

[Out]

2/9*(3*x*log(x) - 2*x)*sqrt(x)

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Sympy [A]  time = 3.09091, size = 66, normalized size = 3.14 \[ \begin{cases} \frac{2 x^{\frac{3}{2}} \log{\left (x \right )}}{3} - \frac{4 x^{\frac{3}{2}}}{9} & \text{for}\: \left |{x}\right | < 1 \\- \frac{2 x^{\frac{3}{2}} \log{\left (\frac{1}{x} \right )}}{3} - \frac{4 x^{\frac{3}{2}}}{9} & \text{for}\: \left |{\frac{1}{x}}\right | < 1 \\-{G_{3, 3}^{2, 1}\left (\begin{matrix} 1 & \frac{5}{2}, \frac{5}{2} \\\frac{3}{2}, \frac{3}{2} & 0 \end{matrix} \middle |{x} \right )} +{G_{3, 3}^{0, 3}\left (\begin{matrix} \frac{5}{2}, \frac{5}{2}, 1 & \\ & \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{x} \right )} & \text{otherwise} \end{cases} \]

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, Abs(1/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/XCAS [A]  time = 0.204378, size = 18, normalized size = 0.86 \[ \frac{2}{3} \, x^{\frac{3}{2}}{\rm ln}\left (x\right ) - \frac{4}{9} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)*log(x),x, algorithm="giac")

[Out]

2/3*x^(3/2)*ln(x) - 4/9*x^(3/2)