### 3.229 $$\int x \log (\sqrt [3]{1+3 x}) \, dx$$

Optimal. Leaf size=40 $-\frac{x^2}{12}+\frac{1}{2} x^2 \log \left (\sqrt [3]{3 x+1}\right )+\frac{x}{18}-\frac{1}{54} \log (3 x+1)$

[Out]

x/18 - x^2/12 + (x^2*Log[(1 + 3*x)^(1/3)])/2 - Log[1 + 3*x]/54

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Rubi [A]  time = 0.0159573, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {2395, 43} $-\frac{x^2}{12}+\frac{1}{2} x^2 \log \left (\sqrt [3]{3 x+1}\right )+\frac{x}{18}-\frac{1}{54} \log (3 x+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Log[(1 + 3*x)^(1/3)],x]

[Out]

x/18 - x^2/12 + (x^2*Log[(1 + 3*x)^(1/3)])/2 - Log[1 + 3*x]/54

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \log \left (\sqrt [3]{1+3 x}\right ) \, dx &=\frac{1}{2} x^2 \log \left (\sqrt [3]{1+3 x}\right )-\frac{1}{2} \int \frac{x^2}{1+3 x} \, dx\\ &=\frac{1}{2} x^2 \log \left (\sqrt [3]{1+3 x}\right )-\frac{1}{2} \int \left (-\frac{1}{9}+\frac{x}{3}+\frac{1}{9 (1+3 x)}\right ) \, dx\\ &=\frac{x}{18}-\frac{x^2}{12}+\frac{1}{2} x^2 \log \left (\sqrt [3]{1+3 x}\right )-\frac{1}{54} \log (1+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0083492, size = 40, normalized size = 1. $\frac{1}{3} \left (-\frac{x^2}{4}+\frac{1}{2} x^2 \log (3 x+1)+\frac{x}{6}-\frac{1}{18} \log (3 x+1)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Log[(1 + 3*x)^(1/3)],x]

[Out]

(x/6 - x^2/4 - Log[1 + 3*x]/18 + (x^2*Log[1 + 3*x])/2)/3

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Maple [A]  time = 0.003, size = 39, normalized size = 1. \begin{align*}{\frac{\ln \left ( 1+3\,x \right ) \left ( 1+3\,x \right ) ^{2}}{54}}-{\frac{{x}^{2}}{12}}+{\frac{x}{18}}+{\frac{1}{36}}-{\frac{ \left ( 1+3\,x \right ) \ln \left ( 1+3\,x \right ) }{27}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*x*ln(1+3*x),x)

[Out]

1/54*ln(1+3*x)*(1+3*x)^2-1/12*x^2+1/18*x+1/36-1/27*(1+3*x)*ln(1+3*x)

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Maxima [A]  time = 1.04281, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{6} \, x^{2} \log \left (3 \, x + 1\right ) - \frac{1}{12} \, x^{2} + \frac{1}{18} \, x - \frac{1}{54} \, \log \left (3 \, x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*log(1+3*x),x, algorithm="maxima")

[Out]

1/6*x^2*log(3*x + 1) - 1/12*x^2 + 1/18*x - 1/54*log(3*x + 1)

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Fricas [A]  time = 1.86306, size = 70, normalized size = 1.75 \begin{align*} -\frac{1}{12} \, x^{2} + \frac{1}{54} \,{\left (9 \, x^{2} - 1\right )} \log \left (3 \, x + 1\right ) + \frac{1}{18} \, x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*log(1+3*x),x, algorithm="fricas")

[Out]

-1/12*x^2 + 1/54*(9*x^2 - 1)*log(3*x + 1) + 1/18*x

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Sympy [A]  time = 0.117948, size = 27, normalized size = 0.68 \begin{align*} \frac{x^{2} \log{\left (3 x + 1 \right )}}{6} - \frac{x^{2}}{12} + \frac{x}{18} - \frac{\log{\left (3 x + 1 \right )}}{54} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*ln(1+3*x),x)

[Out]

x**2*log(3*x + 1)/6 - x**2/12 + x/18 - log(3*x + 1)/54

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Giac [A]  time = 1.21409, size = 57, normalized size = 1.42 \begin{align*} \frac{1}{54} \,{\left (3 \, x + 1\right )}^{2} \log \left (3 \, x + 1\right ) - \frac{1}{108} \,{\left (3 \, x + 1\right )}^{2} - \frac{1}{27} \,{\left (3 \, x + 1\right )} \log \left (3 \, x + 1\right ) + \frac{1}{9} \, x + \frac{1}{27} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*log(1+3*x),x, algorithm="giac")

[Out]

1/54*(3*x + 1)^2*log(3*x + 1) - 1/108*(3*x + 1)^2 - 1/27*(3*x + 1)*log(3*x + 1) + 1/9*x + 1/27