3.85.63 \(\int (4620 x^2+2160 x^3+240 x^4+(816 x^2+192 x^3) \log (x)+36 x^2 \log ^2(x)) \, dx\)

Optimal. Leaf size=18 \[ 12 x (3 x+x (8+2 x+\log (x)))^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 2.61, number of steps used = 8, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 43, 2334, 12, 2305, 2304} \begin {gather*} 48 x^5+528 x^4+1452 x^3+12 x^3 \log ^2(x)-8 x^3 \log (x)+16 \left (3 x^4+17 x^3\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4620*x^2 + 2160*x^3 + 240*x^4 + (816*x^2 + 192*x^3)*Log[x] + 36*x^2*Log[x]^2,x]

[Out]

1452*x^3 + 528*x^4 + 48*x^5 - 8*x^3*Log[x] + 16*(17*x^3 + 3*x^4)*Log[x] + 12*x^3*Log[x]^2

Rule 12

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

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])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=1540 x^3+540 x^4+48 x^5+36 \int x^2 \log ^2(x) \, dx+\int \left (816 x^2+192 x^3\right ) \log (x) \, dx\\ &=1540 x^3+540 x^4+48 x^5+12 x^3 \log ^2(x)-24 \int x^2 \log (x) \, dx+\int x^2 (816+192 x) \log (x) \, dx\\ &=\frac {4628 x^3}{3}+540 x^4+48 x^5-8 x^3 \log (x)+16 \left (17 x^3+3 x^4\right ) \log (x)+12 x^3 \log ^2(x)-\int 16 x^2 (17+3 x) \, dx\\ &=\frac {4628 x^3}{3}+540 x^4+48 x^5-8 x^3 \log (x)+16 \left (17 x^3+3 x^4\right ) \log (x)+12 x^3 \log ^2(x)-16 \int x^2 (17+3 x) \, dx\\ &=\frac {4628 x^3}{3}+540 x^4+48 x^5-8 x^3 \log (x)+16 \left (17 x^3+3 x^4\right ) \log (x)+12 x^3 \log ^2(x)-16 \int \left (17 x^2+3 x^3\right ) \, dx\\ &=1452 x^3+528 x^4+48 x^5-8 x^3 \log (x)+16 \left (17 x^3+3 x^4\right ) \log (x)+12 x^3 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 39, normalized size = 2.17 \begin {gather*} 1452 x^3+528 x^4+48 x^5+264 x^3 \log (x)+48 x^4 \log (x)+12 x^3 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4620*x^2 + 2160*x^3 + 240*x^4 + (816*x^2 + 192*x^3)*Log[x] + 36*x^2*Log[x]^2,x]

[Out]

1452*x^3 + 528*x^4 + 48*x^5 + 264*x^3*Log[x] + 48*x^4*Log[x] + 12*x^3*Log[x]^2

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fricas [B]  time = 0.72, size = 40, normalized size = 2.22 \begin {gather*} 48 \, x^{5} + 12 \, x^{3} \log \relax (x)^{2} + 528 \, x^{4} + 1452 \, x^{3} + 24 \, {\left (2 \, x^{4} + 11 \, x^{3}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(36*x^2*log(x)^2+(192*x^3+816*x^2)*log(x)+240*x^4+2160*x^3+4620*x^2,x, algorithm="fricas")

[Out]

48*x^5 + 12*x^3*log(x)^2 + 528*x^4 + 1452*x^3 + 24*(2*x^4 + 11*x^3)*log(x)

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giac [B]  time = 0.16, size = 39, normalized size = 2.17 \begin {gather*} 48 \, x^{5} + 48 \, x^{4} \log \relax (x) + 12 \, x^{3} \log \relax (x)^{2} + 528 \, x^{4} + 264 \, x^{3} \log \relax (x) + 1452 \, x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(36*x^2*log(x)^2+(192*x^3+816*x^2)*log(x)+240*x^4+2160*x^3+4620*x^2,x, algorithm="giac")

[Out]

48*x^5 + 48*x^4*log(x) + 12*x^3*log(x)^2 + 528*x^4 + 264*x^3*log(x) + 1452*x^3

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maple [B]  time = 0.03, size = 40, normalized size = 2.22

method	result	size
default	\(48 x^{4} \ln \relax (x )+528 x^{4}+264 x^{3} \ln \relax (x )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \relax (x )^{2}\)	\(40\)
norman	\(48 x^{4} \ln \relax (x )+528 x^{4}+264 x^{3} \ln \relax (x )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \relax (x )^{2}\)	\(40\)
risch	\(48 x^{4} \ln \relax (x )+528 x^{4}+264 x^{3} \ln \relax (x )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \relax (x )^{2}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(36*x^2*ln(x)^2+(192*x^3+816*x^2)*ln(x)+240*x^4+2160*x^3+4620*x^2,x,method=_RETURNVERBOSE)

[Out]

48*x^4*ln(x)+528*x^4+264*x^3*ln(x)+1452*x^3+48*x^5+12*x^3*ln(x)^2

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maxima [B]  time = 0.35, size = 48, normalized size = 2.67 \begin {gather*} 48 \, x^{5} + \frac {4}{3} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} + 528 \, x^{4} + \frac {4348}{3} \, x^{3} + 16 \, {\left (3 \, x^{4} + 17 \, x^{3}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(36*x^2*log(x)^2+(192*x^3+816*x^2)*log(x)+240*x^4+2160*x^3+4620*x^2,x, algorithm="maxima")

[Out]

48*x^5 + 4/3*(9*log(x)^2 - 6*log(x) + 2)*x^3 + 528*x^4 + 4348/3*x^3 + 16*(3*x^4 + 17*x^3)*log(x)

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mupad [B]  time = 5.28, size = 14, normalized size = 0.78 \begin {gather*} 12\,x^3\,{\left (2\,x+\ln \relax (x)+11\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)*(816*x^2 + 192*x^3) + 36*x^2*log(x)^2 + 4620*x^2 + 2160*x^3 + 240*x^4,x)

[Out]

12*x^3*(2*x + log(x) + 11)^2

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sympy [B]  time = 0.13, size = 37, normalized size = 2.06 \begin {gather*} 48 x^{5} + 528 x^{4} + 12 x^{3} \log {\relax (x )}^{2} + 1452 x^{3} + \left (48 x^{4} + 264 x^{3}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(36*x**2*ln(x)**2+(192*x**3+816*x**2)*ln(x)+240*x**4+2160*x**3+4620*x**2,x)

[Out]

48*x**5 + 528*x**4 + 12*x**3*log(x)**2 + 1452*x**3 + (48*x**4 + 264*x**3)*log(x)

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