3.51.1 \(\int \frac {324 x^4+195 x^5+26 x^6+x^7+(141 x+24 x^2+x^3+1296 x^4+780 x^5+104 x^6+4 x^7) \log (x)+(324+195 x+26 x^2+x^3) \log (\frac {27+14 x+x^2}{12+x})}{324 x+195 x^2+26 x^3+x^4} \, dx\)

Optimal. Leaf size=20 \[ \log (x) \left (x^4+\log \left (2+\frac {1}{4+\frac {x}{3}}+x\right )\right ) \]

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Rubi [A]  time = 0.57, antiderivative size = 25, normalized size of antiderivative = 1.25, number of steps used = 28, number of rules used = 7, integrand size = 103, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6688, 2357, 2304, 2317, 2391, 14, 2524} \begin {gather*} x^4 \log (x)+\log (x) \log \left (\frac {x^2+14 x+27}{x+12}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(324*x^4 + 195*x^5 + 26*x^6 + x^7 + (141*x + 24*x^2 + x^3 + 1296*x^4 + 780*x^5 + 104*x^6 + 4*x^7)*Log[x] +
(324 + 195*x + 26*x^2 + x^3)*Log[(27 + 14*x + x^2)/(12 + x)])/(324*x + 195*x^2 + 26*x^3 + x^4),x]

[Out]

x^4*Log[x] + Log[x]*Log[(27 + 14*x + x^2)/(12 + x)]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2317

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

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\left (141+24 x+x^2+1296 x^3+780 x^4+104 x^5+4 x^6\right ) \log (x)}{324+195 x+26 x^2+x^3}+\frac {x^4+\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x}\right ) \, dx\\ &=\int \frac {\left (141+24 x+x^2+1296 x^3+780 x^4+104 x^5+4 x^6\right ) \log (x)}{324+195 x+26 x^2+x^3} \, dx+\int \frac {x^4+\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x} \, dx\\ &=\int \left (4 x^3 \log (x)+\frac {\left (141+24 x+x^2\right ) \log (x)}{324+195 x+26 x^2+x^3}\right ) \, dx+\int \left (x^3+\frac {\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x}\right ) \, dx\\ &=\frac {x^4}{4}+4 \int x^3 \log (x) \, dx+\int \frac {\left (141+24 x+x^2\right ) \log (x)}{324+195 x+26 x^2+x^3} \, dx+\int \frac {\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-\int \frac {(12+x) \left (\frac {14+2 x}{12+x}-\frac {27+14 x+x^2}{(12+x)^2}\right ) \log (x)}{27+14 x+x^2} \, dx+\int \left (\frac {\log (x)}{-12-x}+\frac {2 (7+x) \log (x)}{27+14 x+x^2}\right ) \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )+2 \int \frac {(7+x) \log (x)}{27+14 x+x^2} \, dx+\int \frac {\log (x)}{-12-x} \, dx-\int \left (\frac {\log (x)}{-12-x}+\frac {2 (7+x) \log (x)}{27+14 x+x^2}\right ) \, dx\\ &=x^4 \log (x)-\log \left (1+\frac {x}{12}\right ) \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-2 \int \frac {(7+x) \log (x)}{27+14 x+x^2} \, dx+2 \int \left (\frac {\log (x)}{14-2 \sqrt {22}+2 x}+\frac {\log (x)}{14+2 \sqrt {22}+2 x}\right ) \, dx+\int \frac {\log \left (1+\frac {x}{12}\right )}{x} \, dx-\int \frac {\log (x)}{-12-x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-\text {Li}_2\left (-\frac {x}{12}\right )+2 \int \frac {\log (x)}{14-2 \sqrt {22}+2 x} \, dx+2 \int \frac {\log (x)}{14+2 \sqrt {22}+2 x} \, dx-2 \int \left (\frac {\log (x)}{14-2 \sqrt {22}+2 x}+\frac {\log (x)}{14+2 \sqrt {22}+2 x}\right ) \, dx-\int \frac {\log \left (1+\frac {x}{12}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (1+\frac {x}{7-\sqrt {22}}\right )+\log (x) \log \left (1+\frac {x}{7+\sqrt {22}}\right )+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-2 \int \frac {\log (x)}{14-2 \sqrt {22}+2 x} \, dx-2 \int \frac {\log (x)}{14+2 \sqrt {22}+2 x} \, dx-\int \frac {\log \left (1+\frac {2 x}{14-2 \sqrt {22}}\right )}{x} \, dx-\int \frac {\log \left (1+\frac {2 x}{14+2 \sqrt {22}}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )+\text {Li}_2\left (-\frac {x}{7-\sqrt {22}}\right )+\text {Li}_2\left (-\frac {x}{7+\sqrt {22}}\right )+\int \frac {\log \left (1+\frac {2 x}{14-2 \sqrt {22}}\right )}{x} \, dx+\int \frac {\log \left (1+\frac {2 x}{14+2 \sqrt {22}}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 25, normalized size = 1.25 \begin {gather*} x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(324*x^4 + 195*x^5 + 26*x^6 + x^7 + (141*x + 24*x^2 + x^3 + 1296*x^4 + 780*x^5 + 104*x^6 + 4*x^7)*Lo
g[x] + (324 + 195*x + 26*x^2 + x^3)*Log[(27 + 14*x + x^2)/(12 + x)])/(324*x + 195*x^2 + 26*x^3 + x^4),x]

[Out]

x^4*Log[x] + Log[x]*Log[(27 + 14*x + x^2)/(12 + x)]

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fricas [A]  time = 0.73, size = 25, normalized size = 1.25 \begin {gather*} x^{4} \log \relax (x) + \log \relax (x) \log \left (\frac {x^{2} + 14 \, x + 27}{x + 12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+26*x^2+195*x+324)*log((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6+780*x^5+1296*x^4+x^3+24*x^2+141*x)*
log(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^4+26*x^3+195*x^2+324*x),x, algorithm="fricas")

[Out]

x^4*log(x) + log(x)*log((x^2 + 14*x + 27)/(x + 12))

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giac [A]  time = 0.21, size = 27, normalized size = 1.35 \begin {gather*} x^{4} \log \relax (x) + \log \left (x^{2} + 14 \, x + 27\right ) \log \relax (x) - \log \left (x + 12\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+26*x^2+195*x+324)*log((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6+780*x^5+1296*x^4+x^3+24*x^2+141*x)*
log(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^4+26*x^3+195*x^2+324*x),x, algorithm="giac")

[Out]

x^4*log(x) + log(x^2 + 14*x + 27)*log(x) - log(x + 12)*log(x)

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maple [A]  time = 0.14, size = 26, normalized size = 1.30

method	result	size
default	\(\ln \relax (x ) \ln \left (\frac {x^{2}+14 x +27}{x +12}\right )+x^{4} \ln \relax (x )\)	\(26\)
risch	\(\ln \relax (x ) \ln \left (x^{2}+14 x +27\right )-\ln \relax (x ) \ln \left (x +12\right )+x^{4} \ln \relax (x )-\frac {i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{x +12}\right ) \mathrm {csgn}\left (i \left (x^{2}+14 x +27\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )}{2}+\frac {i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{x +12}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )^{2}}{2}+\frac {i \pi \ln \relax (x ) \mathrm {csgn}\left (i \left (x^{2}+14 x +27\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )^{2}}{2}-\frac {i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )^{3}}{2}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+26*x^2+195*x+324)*ln((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6+780*x^5+1296*x^4+x^3+24*x^2+141*x)*ln(x)+x
^7+26*x^6+195*x^5+324*x^4)/(x^4+26*x^3+195*x^2+324*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)*ln((x^2+14*x+27)/(x+12))+x^4*ln(x)

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maxima [A]  time = 0.48, size = 27, normalized size = 1.35 \begin {gather*} x^{4} \log \relax (x) + \log \left (x^{2} + 14 \, x + 27\right ) \log \relax (x) - \log \left (x + 12\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+26*x^2+195*x+324)*log((x^2+14*x+27)/(x+12))+(4*x^7+104*x^6+780*x^5+1296*x^4+x^3+24*x^2+141*x)*
log(x)+x^7+26*x^6+195*x^5+324*x^4)/(x^4+26*x^3+195*x^2+324*x),x, algorithm="maxima")

[Out]

x^4*log(x) + log(x^2 + 14*x + 27)*log(x) - log(x + 12)*log(x)

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mupad [B]  time = 3.37, size = 22, normalized size = 1.10 \begin {gather*} \ln \relax (x)\,\left (\ln \left (\frac {x^2+14\,x+27}{x+12}\right )+x^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(141*x + 24*x^2 + x^3 + 1296*x^4 + 780*x^5 + 104*x^6 + 4*x^7) + log((14*x + x^2 + 27)/(x + 12))*(1
95*x + 26*x^2 + x^3 + 324) + 324*x^4 + 195*x^5 + 26*x^6 + x^7)/(324*x + 195*x^2 + 26*x^3 + x^4),x)

[Out]

log(x)*(log((14*x + x^2 + 27)/(x + 12)) + x^4)

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sympy [A]  time = 0.39, size = 22, normalized size = 1.10 \begin {gather*} x^{4} \log {\relax (x )} + \log {\relax (x )} \log {\left (\frac {x^{2} + 14 x + 27}{x + 12} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+26*x**2+195*x+324)*ln((x**2+14*x+27)/(x+12))+(4*x**7+104*x**6+780*x**5+1296*x**4+x**3+24*x**2
+141*x)*ln(x)+x**7+26*x**6+195*x**5+324*x**4)/(x**4+26*x**3+195*x**2+324*x),x)

[Out]

x**4*log(x) + log(x)*log((x**2 + 14*x + 27)/(x + 12))

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