3.35.77 \(\int \frac {1}{5} (3872+5368 x+648 x^2-144 x^3+(-2904-3696 x-792 x^2) \log (x)+(484+528 x+108 x^2) \log ^2(x)) \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{5} x (2+x+5 (4+x))^2 \left (-x+(-4+\log (x))^2\right ) \]

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Rubi [B]  time = 0.09, antiderivative size = 81, normalized size of antiderivative = 3.24, number of steps used = 15, number of rules used = 6, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {12, 2356, 2295, 2304, 2296, 2305} \begin {gather*} -\frac {36 x^4}{5}+\frac {312 x^3}{5}+\frac {36}{5} x^3 \log ^2(x)-\frac {288}{5} x^3 \log (x)+748 x^2+\frac {264}{5} x^2 \log ^2(x)-\frac {2112}{5} x^2 \log (x)+\frac {7744 x}{5}+\frac {484}{5} x \log ^2(x)-\frac {3872}{5} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3872 + 5368*x + 648*x^2 - 144*x^3 + (-2904 - 3696*x - 792*x^2)*Log[x] + (484 + 528*x + 108*x^2)*Log[x]^2)
/5,x]

[Out]

(7744*x)/5 + 748*x^2 + (312*x^3)/5 - (36*x^4)/5 - (3872*x*Log[x])/5 - (2112*x^2*Log[x])/5 - (288*x^3*Log[x])/5
+ (484*x*Log[x]^2)/5 + (264*x^2*Log[x]^2)/5 + (36*x^3*Log[x]^2)/5

Rule 12

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*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} &=\frac {1}{5} \int \left (3872+5368 x+648 x^2-144 x^3+\left (-2904-3696 x-792 x^2\right ) \log (x)+\left (484+528 x+108 x^2\right ) \log ^2(x)\right ) \, dx\\ &=\frac {3872 x}{5}+\frac {2684 x^2}{5}+\frac {216 x^3}{5}-\frac {36 x^4}{5}+\frac {1}{5} \int \left (-2904-3696 x-792 x^2\right ) \log (x) \, dx+\frac {1}{5} \int \left (484+528 x+108 x^2\right ) \log ^2(x) \, dx\\ &=\frac {3872 x}{5}+\frac {2684 x^2}{5}+\frac {216 x^3}{5}-\frac {36 x^4}{5}+\frac {1}{5} \int \left (-2904 \log (x)-3696 x \log (x)-792 x^2 \log (x)\right ) \, dx+\frac {1}{5} \int \left (484 \log ^2(x)+528 x \log ^2(x)+108 x^2 \log ^2(x)\right ) \, dx\\ &=\frac {3872 x}{5}+\frac {2684 x^2}{5}+\frac {216 x^3}{5}-\frac {36 x^4}{5}+\frac {108}{5} \int x^2 \log ^2(x) \, dx+\frac {484}{5} \int \log ^2(x) \, dx+\frac {528}{5} \int x \log ^2(x) \, dx-\frac {792}{5} \int x^2 \log (x) \, dx-\frac {2904}{5} \int \log (x) \, dx-\frac {3696}{5} \int x \log (x) \, dx\\ &=\frac {6776 x}{5}+\frac {3608 x^2}{5}+\frac {304 x^3}{5}-\frac {36 x^4}{5}-\frac {2904}{5} x \log (x)-\frac {1848}{5} x^2 \log (x)-\frac {264}{5} x^3 \log (x)+\frac {484}{5} x \log ^2(x)+\frac {264}{5} x^2 \log ^2(x)+\frac {36}{5} x^3 \log ^2(x)-\frac {72}{5} \int x^2 \log (x) \, dx-\frac {528}{5} \int x \log (x) \, dx-\frac {968}{5} \int \log (x) \, dx\\ &=\frac {7744 x}{5}+748 x^2+\frac {312 x^3}{5}-\frac {36 x^4}{5}-\frac {3872}{5} x \log (x)-\frac {2112}{5} x^2 \log (x)-\frac {288}{5} x^3 \log (x)+\frac {484}{5} x \log ^2(x)+\frac {264}{5} x^2 \log ^2(x)+\frac {36}{5} x^3 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 67, normalized size = 2.68 \begin {gather*} -\frac {4}{5} \left (-1936 x-935 x^2-78 x^3+9 x^4+968 x \log (x)+528 x^2 \log (x)+72 x^3 \log (x)-121 x \log ^2(x)-66 x^2 \log ^2(x)-9 x^3 \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3872 + 5368*x + 648*x^2 - 144*x^3 + (-2904 - 3696*x - 792*x^2)*Log[x] + (484 + 528*x + 108*x^2)*Log
[x]^2)/5,x]

[Out]

(-4*(-1936*x - 935*x^2 - 78*x^3 + 9*x^4 + 968*x*Log[x] + 528*x^2*Log[x] + 72*x^3*Log[x] - 121*x*Log[x]^2 - 66*
x^2*Log[x]^2 - 9*x^3*Log[x]^2))/5

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fricas [B]  time = 0.92, size = 57, normalized size = 2.28 \begin {gather*} -\frac {36}{5} \, x^{4} + \frac {312}{5} \, x^{3} + \frac {4}{5} \, {\left (9 \, x^{3} + 66 \, x^{2} + 121 \, x\right )} \log \relax (x)^{2} + 748 \, x^{2} - \frac {32}{5} \, {\left (9 \, x^{3} + 66 \, x^{2} + 121 \, x\right )} \log \relax (x) + \frac {7744}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(108*x^2+528*x+484)*log(x)^2+1/5*(-792*x^2-3696*x-2904)*log(x)-144/5*x^3+648/5*x^2+5368/5*x+3872
/5,x, algorithm="fricas")

[Out]

-36/5*x^4 + 312/5*x^3 + 4/5*(9*x^3 + 66*x^2 + 121*x)*log(x)^2 + 748*x^2 - 32/5*(9*x^3 + 66*x^2 + 121*x)*log(x)
+ 7744/5*x

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giac [B]  time = 0.23, size = 63, normalized size = 2.52 \begin {gather*} \frac {36}{5} \, x^{3} \log \relax (x)^{2} - \frac {36}{5} \, x^{4} - \frac {288}{5} \, x^{3} \log \relax (x) + \frac {264}{5} \, x^{2} \log \relax (x)^{2} + \frac {312}{5} \, x^{3} - \frac {2112}{5} \, x^{2} \log \relax (x) + \frac {484}{5} \, x \log \relax (x)^{2} + 748 \, x^{2} - \frac {3872}{5} \, x \log \relax (x) + \frac {7744}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(108*x^2+528*x+484)*log(x)^2+1/5*(-792*x^2-3696*x-2904)*log(x)-144/5*x^3+648/5*x^2+5368/5*x+3872
/5,x, algorithm="giac")

[Out]

36/5*x^3*log(x)^2 - 36/5*x^4 - 288/5*x^3*log(x) + 264/5*x^2*log(x)^2 + 312/5*x^3 - 2112/5*x^2*log(x) + 484/5*x
*log(x)^2 + 748*x^2 - 3872/5*x*log(x) + 7744/5*x

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maple [B]  time = 0.02, size = 64, normalized size = 2.56

method	result	size
default	\(\frac {7744 x}{5}+748 x^{2}+\frac {312 x^{3}}{5}-\frac {36 x^{4}}{5}-\frac {288 x^{3} \ln \relax (x )}{5}-\frac {2112 x^{2} \ln \relax (x )}{5}-\frac {3872 x \ln \relax (x )}{5}+\frac {36 x^{3} \ln \relax (x )^{2}}{5}+\frac {264 x^{2} \ln \relax (x )^{2}}{5}+\frac {484 x \ln \relax (x )^{2}}{5}\)	\(64\)
norman	\(\frac {7744 x}{5}+748 x^{2}+\frac {312 x^{3}}{5}-\frac {36 x^{4}}{5}-\frac {288 x^{3} \ln \relax (x )}{5}-\frac {2112 x^{2} \ln \relax (x )}{5}-\frac {3872 x \ln \relax (x )}{5}+\frac {36 x^{3} \ln \relax (x )^{2}}{5}+\frac {264 x^{2} \ln \relax (x )^{2}}{5}+\frac {484 x \ln \relax (x )^{2}}{5}\)	\(64\)
risch	\(\frac {\left (36 x^{3}+264 x^{2}+484 x \right ) \ln \relax (x )^{2}}{5}+\frac {\left (-24 x^{3}-264 x^{2}-968 x \right ) \ln \relax (x )}{5}+\frac {312 x^{3}}{5}+748 x^{2}+\frac {7744 x}{5}+\frac {\left (-264 x^{3}-1848 x^{2}-2904 x \right ) \ln \relax (x )}{5}-\frac {36 x^{4}}{5}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(108*x^2+528*x+484)*ln(x)^2+1/5*(-792*x^2-3696*x-2904)*ln(x)-144/5*x^3+648/5*x^2+5368/5*x+3872/5,x,met
hod=_RETURNVERBOSE)

[Out]

7744/5*x+748*x^2+312/5*x^3-36/5*x^4-288/5*x^3*ln(x)-2112/5*x^2*ln(x)-3872/5*x*ln(x)+36/5*x^3*ln(x)^2+264/5*x^2
*ln(x)^2+484/5*x*ln(x)^2

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maxima [B]  time = 0.41, size = 82, normalized size = 3.28 \begin {gather*} \frac {4}{5} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} - \frac {36}{5} \, x^{4} + \frac {132}{5} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + \frac {304}{5} \, x^{3} + \frac {484}{5} \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + \frac {3608}{5} \, x^{2} - \frac {264}{5} \, {\left (x^{3} + 7 \, x^{2} + 11 \, x\right )} \log \relax (x) + \frac {6776}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(108*x^2+528*x+484)*log(x)^2+1/5*(-792*x^2-3696*x-2904)*log(x)-144/5*x^3+648/5*x^2+5368/5*x+3872
/5,x, algorithm="maxima")

[Out]

4/5*(9*log(x)^2 - 6*log(x) + 2)*x^3 - 36/5*x^4 + 132/5*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 304/5*x^3 + 484/5*(lo
g(x)^2 - 2*log(x) + 2)*x + 3608/5*x^2 - 264/5*(x^3 + 7*x^2 + 11*x)*log(x) + 6776/5*x

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mupad [B]  time = 2.04, size = 23, normalized size = 0.92 \begin {gather*} -\frac {4\,x\,{\left (3\,x+11\right )}^2\,\left (-{\ln \relax (x)}^2+8\,\ln \relax (x)+x-16\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5368*x)/5 + (log(x)^2*(528*x + 108*x^2 + 484))/5 - (log(x)*(3696*x + 792*x^2 + 2904))/5 + (648*x^2)/5 - (
144*x^3)/5 + 3872/5,x)

[Out]

-(4*x*(3*x + 11)^2*(x + 8*log(x) - log(x)^2 - 16))/5

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sympy [B]  time = 0.17, size = 70, normalized size = 2.80 \begin {gather*} - \frac {36 x^{4}}{5} + \frac {312 x^{3}}{5} + 748 x^{2} + \frac {7744 x}{5} + \left (- \frac {288 x^{3}}{5} - \frac {2112 x^{2}}{5} - \frac {3872 x}{5}\right ) \log {\relax (x )} + \left (\frac {36 x^{3}}{5} + \frac {264 x^{2}}{5} + \frac {484 x}{5}\right ) \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(108*x**2+528*x+484)*ln(x)**2+1/5*(-792*x**2-3696*x-2904)*ln(x)-144/5*x**3+648/5*x**2+5368/5*x+3
872/5,x)

[Out]

-36*x**4/5 + 312*x**3/5 + 748*x**2 + 7744*x/5 + (-288*x**3/5 - 2112*x**2/5 - 3872*x/5)*log(x) + (36*x**3/5 + 2
64*x**2/5 + 484*x/5)*log(x)**2

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