3.5.30 \(\int (-432 x^2-1800 x^3+1080 x^4+1944 x^5+(-648 x+648 x^2+2592 x^3) \log (x)+648 x \log ^2(x)) \, dx\)

Optimal. Leaf size=18 \[ 36 x^2 \left (-3+x+3 \left (x^2+\log (x)\right )\right )^2 \]

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Rubi [B]  time = 0.12, antiderivative size = 56, normalized size of antiderivative = 3.11, number of steps used = 9, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {1594, 2356, 2304, 2305} \begin {gather*} 324 x^6+216 x^5-612 x^4+648 x^4 \log (x)-216 x^3+216 x^3 \log (x)+324 x^2+324 x^2 \log ^2(x)-648 x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-432*x^2 - 1800*x^3 + 1080*x^4 + 1944*x^5 + (-648*x + 648*x^2 + 2592*x^3)*Log[x] + 648*x*Log[x]^2,x]

[Out]

324*x^2 - 216*x^3 - 612*x^4 + 216*x^5 + 324*x^6 - 648*x^2*Log[x] + 216*x^3*Log[x] + 648*x^4*Log[x] + 324*x^2*L
og[x]^2

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 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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} &=-144 x^3-450 x^4+216 x^5+324 x^6+648 \int x \log ^2(x) \, dx+\int \left (-648 x+648 x^2+2592 x^3\right ) \log (x) \, dx\\ &=-144 x^3-450 x^4+216 x^5+324 x^6+324 x^2 \log ^2(x)-648 \int x \log (x) \, dx+\int x \left (-648+648 x+2592 x^2\right ) \log (x) \, dx\\ &=162 x^2-144 x^3-450 x^4+216 x^5+324 x^6-324 x^2 \log (x)+324 x^2 \log ^2(x)+\int \left (-648 x \log (x)+648 x^2 \log (x)+2592 x^3 \log (x)\right ) \, dx\\ &=162 x^2-144 x^3-450 x^4+216 x^5+324 x^6-324 x^2 \log (x)+324 x^2 \log ^2(x)-648 \int x \log (x) \, dx+648 \int x^2 \log (x) \, dx+2592 \int x^3 \log (x) \, dx\\ &=324 x^2-216 x^3-612 x^4+216 x^5+324 x^6-648 x^2 \log (x)+216 x^3 \log (x)+648 x^4 \log (x)+324 x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 56, normalized size = 3.11 \begin {gather*} 324 x^2-216 x^3-612 x^4+216 x^5+324 x^6-648 x^2 \log (x)+216 x^3 \log (x)+648 x^4 \log (x)+324 x^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-432*x^2 - 1800*x^3 + 1080*x^4 + 1944*x^5 + (-648*x + 648*x^2 + 2592*x^3)*Log[x] + 648*x*Log[x]^2,x]

[Out]

324*x^2 - 216*x^3 - 612*x^4 + 216*x^5 + 324*x^6 - 648*x^2*Log[x] + 216*x^3*Log[x] + 648*x^4*Log[x] + 324*x^2*L
og[x]^2

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fricas [B]  time = 0.94, size = 53, normalized size = 2.94 \begin {gather*} 324 \, x^{6} + 216 \, x^{5} - 612 \, x^{4} + 324 \, x^{2} \log \relax (x)^{2} - 216 \, x^{3} + 324 \, x^{2} + 216 \, {\left (3 \, x^{4} + x^{3} - 3 \, x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(648*x*log(x)^2+(2592*x^3+648*x^2-648*x)*log(x)+1944*x^5+1080*x^4-1800*x^3-432*x^2,x, algorithm="fric
as")

[Out]

324*x^6 + 216*x^5 - 612*x^4 + 324*x^2*log(x)^2 - 216*x^3 + 324*x^2 + 216*(3*x^4 + x^3 - 3*x^2)*log(x)

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giac [B]  time = 0.36, size = 56, normalized size = 3.11 \begin {gather*} 324 \, x^{6} + 216 \, x^{5} + 648 \, x^{4} \log \relax (x) - 612 \, x^{4} + 216 \, x^{3} \log \relax (x) + 324 \, x^{2} \log \relax (x)^{2} - 216 \, x^{3} - 648 \, x^{2} \log \relax (x) + 324 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(648*x*log(x)^2+(2592*x^3+648*x^2-648*x)*log(x)+1944*x^5+1080*x^4-1800*x^3-432*x^2,x, algorithm="giac
")

[Out]

324*x^6 + 216*x^5 + 648*x^4*log(x) - 612*x^4 + 216*x^3*log(x) + 324*x^2*log(x)^2 - 216*x^3 - 648*x^2*log(x) +
324*x^2

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maple [B]  time = 0.02, size = 57, normalized size = 3.17

method	result	size
default	\(648 x^{4} \ln \relax (x )-612 x^{4}+216 x^{3} \ln \relax (x )-216 x^{3}-648 x^{2} \ln \relax (x )+324 x^{2}+216 x^{5}+324 x^{6}+324 x^{2} \ln \relax (x )^{2}\)	\(57\)
norman	\(648 x^{4} \ln \relax (x )-612 x^{4}+216 x^{3} \ln \relax (x )-216 x^{3}-648 x^{2} \ln \relax (x )+324 x^{2}+216 x^{5}+324 x^{6}+324 x^{2} \ln \relax (x )^{2}\)	\(57\)
risch	\(648 x^{4} \ln \relax (x )-612 x^{4}+216 x^{3} \ln \relax (x )-216 x^{3}-648 x^{2} \ln \relax (x )+324 x^{2}+216 x^{5}+324 x^{6}+324 x^{2} \ln \relax (x )^{2}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(648*x*ln(x)^2+(2592*x^3+648*x^2-648*x)*ln(x)+1944*x^5+1080*x^4-1800*x^3-432*x^2,x,method=_RETURNVERBOSE)

[Out]

648*x^4*ln(x)-612*x^4+216*x^3*ln(x)-216*x^3-648*x^2*ln(x)+324*x^2+216*x^5+324*x^6+324*x^2*ln(x)^2

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maxima [B]  time = 0.59, size = 63, normalized size = 3.50 \begin {gather*} 324 \, x^{6} + 216 \, x^{5} - 612 \, x^{4} + 162 \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} - 216 \, x^{3} + 162 \, x^{2} + 108 \, {\left (6 \, x^{4} + 2 \, x^{3} - 3 \, x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(648*x*log(x)^2+(2592*x^3+648*x^2-648*x)*log(x)+1944*x^5+1080*x^4-1800*x^3-432*x^2,x, algorithm="maxi
ma")

[Out]

324*x^6 + 216*x^5 - 612*x^4 + 162*(2*log(x)^2 - 2*log(x) + 1)*x^2 - 216*x^3 + 162*x^2 + 108*(6*x^4 + 2*x^3 - 3
*x^2)*log(x)

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mupad [B]  time = 0.54, size = 19, normalized size = 1.06 \begin {gather*} 36\,x^2\,{\left (x+3\,\ln \relax (x)+3\,x^2-3\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(648*x*log(x)^2 - 432*x^2 - 1800*x^3 + 1080*x^4 + 1944*x^5 + log(x)*(648*x^2 - 648*x + 2592*x^3),x)

[Out]

36*x^2*(x + 3*log(x) + 3*x^2 - 3)^2

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sympy [B]  time = 0.13, size = 53, normalized size = 2.94 \begin {gather*} 324 x^{6} + 216 x^{5} - 612 x^{4} - 216 x^{3} + 324 x^{2} \log {\relax (x )}^{2} + 324 x^{2} + \left (648 x^{4} + 216 x^{3} - 648 x^{2}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(648*x*ln(x)**2+(2592*x**3+648*x**2-648*x)*ln(x)+1944*x**5+1080*x**4-1800*x**3-432*x**2,x)

[Out]

324*x**6 + 216*x**5 - 612*x**4 - 216*x**3 + 324*x**2*log(x)**2 + 324*x**2 + (648*x**4 + 216*x**3 - 648*x**2)*l
og(x)

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