### 3.224 $$\int x \log (2-3 x^2) \, dx$$

Optimal. Leaf size=27 $-\frac{x^2}{2}-\frac{1}{6} \left (2-3 x^2\right ) \log \left (2-3 x^2\right )$

[Out]

-x^2/2 - ((2 - 3*x^2)*Log[2 - 3*x^2])/6

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Rubi [A]  time = 0.0212735, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {2454, 2389, 2295} $-\frac{x^2}{2}-\frac{1}{6} \left (2-3 x^2\right ) \log \left (2-3 x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Log[2 - 3*x^2],x]

[Out]

-x^2/2 - ((2 - 3*x^2)*Log[2 - 3*x^2])/6

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int x \log \left (2-3 x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \log (2-3 x) \, dx,x,x^2\right )\\ &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \log (x) \, dx,x,2-3 x^2\right )\right )\\ &=-\frac{x^2}{2}-\frac{1}{6} \left (2-3 x^2\right ) \log \left (2-3 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0037477, size = 26, normalized size = 0.96 $\frac{1}{6} \left (\left (3 x^2-2\right ) \log \left (2-3 x^2\right )-3 x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Log[2 - 3*x^2],x]

[Out]

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

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Maple [A]  time = 0.001, size = 25, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -3\,{x}^{2}+2 \right ) \ln \left ( -3\,{x}^{2}+2 \right ) }{6}}-{\frac{{x}^{2}}{2}}+{\frac{1}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(-3*x^2+2),x)

[Out]

-1/6*(-3*x^2+2)*ln(-3*x^2+2)-1/2*x^2+1/3

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Maxima [A]  time = 1.06472, size = 32, normalized size = 1.19 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{6} \,{\left (3 \, x^{2} - 2\right )} \log \left (-3 \, x^{2} + 2\right ) + \frac{1}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(-3*x^2+2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.84247, size = 59, normalized size = 2.19 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{6} \,{\left (3 \, x^{2} - 2\right )} \log \left (-3 \, x^{2} + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(-3*x^2+2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.126375, size = 27, normalized size = 1. \begin{align*} \frac{x^{2} \log{\left (2 - 3 x^{2} \right )}}{2} - \frac{x^{2}}{2} - \frac{\log{\left (3 x^{2} - 2 \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(-3*x**2+2),x)

[Out]

x**2*log(2 - 3*x**2)/2 - x**2/2 - log(3*x**2 - 2)/3

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Giac [A]  time = 1.33089, size = 32, normalized size = 1.19 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{6} \,{\left (3 \, x^{2} - 2\right )} \log \left (-3 \, x^{2} + 2\right ) + \frac{1}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(-3*x^2+2),x, algorithm="giac")

[Out]

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