3.73.53 $\int \frac{1}{3}(-4x+12x^2+4x^3+(-8x+36x^2+16x^3)\log (x))dx$

Optimal. Leaf size=17 $$\frac{4}{3}{x}^2(-1+3x+x^2)\log (x)$$

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Rubi [A]  time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.53, number of steps used = 8, number of rules used = 4, integrand size = 35, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.114, Rules used = {12, 1594, 2356, 2304} $$\begin{array}{c}\frac{4}{3}{x}^4\log (x)+4x^3\log (x)-\frac{4}{3}{x}^2\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-4*x + 12*x^2 + 4*x^3 + (-8*x + 36*x^2 + 16*x^3)*Log[x])/3,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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
]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int (-4x+12x^2+4x^3+(-8x+36x^2+16x^3)\log (x))dx\\ & =-\frac{2x^2}{3}+\frac{4x^3}{3}+\frac{x^4}{3}+\frac{1}{3}\int (-8x+36x^2+16x^3)\log (x)dx\\ & =-\frac{2x^2}{3}+\frac{4x^3}{3}+\frac{x^4}{3}+\frac{1}{3}\int x(-8+36x+16x^2)\log (x)dx\\ & =-\frac{2x^2}{3}+\frac{4x^3}{3}+\frac{x^4}{3}+\frac{1}{3}\int (-8x\log (x)+36x^2\log (x)+16x^3\log (x))dx\\ & =-\frac{2x^2}{3}+\frac{4x^3}{3}+\frac{x^4}{3}-\frac{8}{3}\int x\log (x)dx+\frac{16}{3}\int x^3\log (x)dx+12\int x^2\log (x)dx\\ & =-\frac{4}{3}{x}^2\log (x)+4x^3\log (x)+\frac{4}{3}{x}^4\log (x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.53 $$\begin{array}{c}-\frac{4}{3}{x}^2\log (x)+4x^3\log (x)+\frac{4}{3}{x}^4\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + 12*x^2 + 4*x^3 + (-8*x + 36*x^2 + 16*x^3)*Log[x])/3,x]

[Out]

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

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fricas [A]  time = 0.70, size = 18, normalized size = 1.06 $$\begin{array}{c}\frac{4}{3}(x^4+3x^3-x^2)\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x^3+36*x^2-8*x)*log(x)+4/3*x^3+4*x^2-4/3*x,x, algorithm="fricas")

[Out]

4/3*(x^4 + 3*x^3 - x^2)*log(x)

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giac [A]  time = 0.14, size = 22, normalized size = 1.29 $$\begin{array}{c}\frac{4}{3}{x}^4\log (x)+4x^3\log (x)-\frac{4}{3}{x}^2\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x^3+36*x^2-8*x)*log(x)+4/3*x^3+4*x^2-4/3*x,x, algorithm="giac")

[Out]

4/3*x^4*log(x) + 4*x^3*log(x) - 4/3*x^2*log(x)

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maple [A]  time = 0.02, size = 16, normalized size = 0.94

method	result	size
risch	$\frac{4x^2(x^2+3x-1)\ln (x)}{3}$	$16$
default	$\frac{4x^4\ln (x)}{3}+4x^3\ln (x)-\frac{4x^2\ln (x)}{3}$	$23$
norman	$\frac{4x^4\ln (x)}{3}+4x^3\ln (x)-\frac{4x^2\ln (x)}{3}$	$23$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(16*x^3+36*x^2-8*x)*ln(x)+4/3*x^3+4*x^2-4/3*x,x,method=_RETURNVERBOSE)

[Out]

4/3*x^2*(x^2+3*x-1)*ln(x)

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maxima [A]  time = 0.35, size = 18, normalized size = 1.06 $$\begin{array}{c}\frac{4}{3}(x^4+3x^3-x^2)\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x^3+36*x^2-8*x)*log(x)+4/3*x^3+4*x^2-4/3*x,x, algorithm="maxima")

[Out]

4/3*(x^4 + 3*x^3 - x^2)*log(x)

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mupad [B]  time = 4.37, size = 15, normalized size = 0.88 $$\begin{array}{c}\frac{4x^2\ln (x)(x^2+3x-1)}{3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x^2 - (4*x)/3 + (4*x^3)/3 + (log(x)*(36*x^2 - 8*x + 16*x^3))/3,x)

[Out]

(4*x^2*log(x)*(3*x + x^2 - 1))/3

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sympy [A]  time = 0.16, size = 20, normalized size = 1.18 $$\begin{array}{c}(\frac{4x^4}{3}+4x^3-\frac{4x^2}{3})\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x**3+36*x**2-8*x)*ln(x)+4/3*x**3+4*x**2-4/3*x,x)

[Out]

(4*x**4/3 + 4*x**3 - 4*x**2/3)*log(x)

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