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

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

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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 {gather*} \frac {4}{3} x^4 \log (x)+4 x^3 \log (x)-\frac {4}{3} x^2 \log (x) \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-4 x+12 x^2+4 x^3+\left (-8 x+36 x^2+16 x^3\right ) \log (x)\right ) \, dx\\ &=-\frac {2 x^2}{3}+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {1}{3} \int \left (-8 x+36 x^2+16 x^3\right ) \log (x) \, dx\\ &=-\frac {2 x^2}{3}+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {1}{3} \int x \left (-8+36 x+16 x^2\right ) \log (x) \, dx\\ &=-\frac {2 x^2}{3}+\frac {4 x^3}{3}+\frac {x^4}{3}+\frac {1}{3} \int \left (-8 x \log (x)+36 x^2 \log (x)+16 x^3 \log (x)\right ) \, dx\\ &=-\frac {2 x^2}{3}+\frac {4 x^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)+4 x^3 \log (x)+\frac {4}{3} x^4 \log (x)\\ \end {aligned} \end {gather*}

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

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 {gather*} \frac {4}{3} \, {\left (x^{4} + 3 \, x^{3} - x^{2}\right )} \log \relax (x) \end {gather*}

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 {gather*} \frac {4}{3} \, x^{4} \log \relax (x) + 4 \, x^{3} \log \relax (x) - \frac {4}{3} \, x^{2} \log \relax (x) \end {gather*}

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	\(\frac {4 x^{2} \left (x^{2}+3 x -1\right ) \ln \relax (x )}{3}\)	\(16\)
default	\(\frac {4 x^{4} \ln \relax (x )}{3}+4 x^{3} \ln \relax (x )-\frac {4 x^{2} \ln \relax (x )}{3}\)	\(23\)
norman	\(\frac {4 x^{4} \ln \relax (x )}{3}+4 x^{3} \ln \relax (x )-\frac {4 x^{2} \ln \relax (x )}{3}\)	\(23\)

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 {gather*} \frac {4}{3} \, {\left (x^{4} + 3 \, x^{3} - x^{2}\right )} \log \relax (x) \end {gather*}

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 {gather*} \frac {4\,x^2\,\ln \relax (x)\,\left (x^2+3\,x-1\right )}{3} \end {gather*}

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 {gather*} \left (\frac {4 x^{4}}{3} + 4 x^{3} - \frac {4 x^{2}}{3}\right ) \log {\relax (x )} \end {gather*}

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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