3.37 $$\int \sqrt{t} \log (t) \, dt$$

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

[Out]

(-4*t^(3/2))/9 + (2*t^(3/2)*Log[t])/3

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Rubi [A]  time = 0.0078074, 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, Rules used = {2304} $\frac{2}{3} t^{3/2} \log (t)-\frac{4 t^{3/2}}{9}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[t]*Log[t],t]

[Out]

(-4*t^(3/2))/9 + (2*t^(3/2)*Log[t])/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{t} \log (t) \, dt &=-\frac{4 t^{3/2}}{9}+\frac{2}{3} t^{3/2} \log (t)\\ \end{align*}

Mathematica [A]  time = 0.0028059, size = 15, normalized size = 0.71 $\frac{2}{9} t^{3/2} (3 \log (t)-2)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[t]*Log[t],t]

[Out]

(2*t^(3/2)*(-2 + 3*Log[t]))/9

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \begin{align*} -{\frac{4}{9}{t}^{{\frac{3}{2}}}}+{\frac{2\,\ln \left ( t \right ) }{3}{t}^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.934268, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3} \, t^{\frac{3}{2}} \log \left (t\right ) - \frac{4}{9} \, t^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*t^(3/2)*log(t) - 4/9*t^(3/2)

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Fricas [A]  time = 2.17516, size = 43, normalized size = 2.05 \begin{align*} \frac{2}{9} \,{\left (3 \, t \log \left (t\right ) - 2 \, t\right )} \sqrt{t} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(3*t*log(t) - 2*t)*sqrt(t)

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Sympy [A]  time = 2.037, size = 66, normalized size = 3.14 \begin{align*} \begin{cases} \frac{2 t^{\frac{3}{2}} \log{\left (t \right )}}{3} - \frac{4 t^{\frac{3}{2}}}{9} & \text{for}\: \left |{t}\right | < 1 \\- \frac{2 t^{\frac{3}{2}} \log{\left (\frac{1}{t} \right )}}{3} - \frac{4 t^{\frac{3}{2}}}{9} & \text{for}\: \frac{1}{\left |{t}\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 |{t} \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 |{t} \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04944, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3} \, t^{\frac{3}{2}} \log \left (t\right ) - \frac{4}{9} \, t^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*t^(3/2)*log(t) - 4/9*t^(3/2)