3.65.56 \(\int \frac {6+39 x+6 x^2+(76+62 x+12 x^2) \log (\frac {2+x}{3})+\log (2 x) (2 x+(4+2 x) \log (\frac {2+x}{3}))}{12+6 x} \, dx\)

Optimal. Leaf size=30 \[ \frac {x}{2}+\left (x+x \left (5+x+\frac {1}{3} \log (2 x)\right )\right ) \log \left (\frac {2+x}{3}\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.37, antiderivative size = 95, normalized size of antiderivative = 3.17, number of steps used = 24, number of rules used = 13, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6742, 2295, 2395, 43, 2389, 2370, 2411, 2351, 2316, 2315, 698, 2317, 2391} \begin {gather*} \frac {x}{2}+\frac {1}{36} (6 x+19)^2 \log \left (\frac {x}{3}+\frac {2}{3}\right )+\frac {2}{3} \log \left (\frac {3}{2}\right ) \log (x)-\frac {2}{3} \log \left (\frac {x}{2}+1\right ) \log (2 x)-\frac {1}{3} (x+2) \log \left (\frac {x+2}{3}\right )+\frac {1}{3} (x+2) \log (2 x) \log \left (\frac {x+2}{3}\right )-\frac {337}{36} \log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + 39*x + 6*x^2 + (76 + 62*x + 12*x^2)*Log[(2 + x)/3] + Log[2*x]*(2*x + (4 + 2*x)*Log[(2 + x)/3]))/(12 +
6*x),x]

[Out]

x/2 + ((19 + 6*x)^2*Log[2/3 + x/3])/36 + (2*Log[3/2]*Log[x])/3 - (2*Log[1 + x/2]*Log[2*x])/3 - ((2 + x)*Log[(2
+ x)/3])/3 + ((2 + x)*Log[2*x]*Log[(2 + x)/3])/3 - (337*Log[2 + x])/36

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 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

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

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

Rule 2389

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

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 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{3} \log \left (\frac {2}{3}+\frac {x}{3}\right ) (19+6 x+\log (2 x))+\frac {6+39 x+6 x^2+2 x \log (2 x)}{6 (2+x)}\right ) \, dx\\ &=\frac {1}{6} \int \frac {6+39 x+6 x^2+2 x \log (2 x)}{2+x} \, dx+\frac {1}{3} \int \log \left (\frac {2}{3}+\frac {x}{3}\right ) (19+6 x+\log (2 x)) \, dx\\ &=\frac {1}{6} \int \left (\frac {3 \left (2+13 x+2 x^2\right )}{2+x}+\frac {2 x \log (2 x)}{2+x}\right ) \, dx+\frac {1}{3} \int \left ((19+6 x) \log \left (\frac {2}{3}+\frac {x}{3}\right )+\log \left (\frac {2}{3}+\frac {x}{3}\right ) \log (2 x)\right ) \, dx\\ &=\frac {1}{3} \int (19+6 x) \log \left (\frac {2}{3}+\frac {x}{3}\right ) \, dx+\frac {1}{3} \int \frac {x \log (2 x)}{2+x} \, dx+\frac {1}{3} \int \log \left (\frac {2}{3}+\frac {x}{3}\right ) \log (2 x) \, dx+\frac {1}{2} \int \frac {2+13 x+2 x^2}{2+x} \, dx\\ &=\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {1}{3} x \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {1}{108} \int \frac {(19+6 x)^2}{\frac {2}{3}+\frac {x}{3}} \, dx+\frac {1}{3} \int \left (\log (2 x)-\frac {2 \log (2 x)}{2+x}\right ) \, dx-\frac {1}{3} \int \left (-1+\frac {(2+x) \log \left (\frac {2+x}{3}\right )}{x}\right ) \, dx+\frac {1}{2} \int \left (9+2 x-\frac {16}{2+x}\right ) \, dx\\ &=\frac {29 x}{6}+\frac {x^2}{2}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {1}{3} x \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-8 \log (2+x)-\frac {1}{108} \int \left (468+108 x+\frac {147}{2+x}\right ) \, dx+\frac {1}{3} \int \log (2 x) \, dx-\frac {1}{3} \int \frac {(2+x) \log \left (\frac {2+x}{3}\right )}{x} \, dx-\frac {2}{3} \int \frac {\log (2 x)}{2+x} \, dx\\ &=\frac {x}{6}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x \log \left (\frac {x}{3}\right )}{-2+x} \, dx,x,2+x\right )+\frac {2}{3} \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx\\ &=\frac {x}{6}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {2 \text {Li}_2\left (-\frac {x}{2}\right )}{3}-\frac {1}{3} \operatorname {Subst}\left (\int \left (\log \left (\frac {x}{3}\right )+\frac {2 \log \left (\frac {x}{3}\right )}{-2+x}\right ) \, dx,x,2+x\right )\\ &=\frac {x}{6}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {2 \text {Li}_2\left (-\frac {x}{2}\right )}{3}-\frac {1}{3} \operatorname {Subst}\left (\int \log \left (\frac {x}{3}\right ) \, dx,x,2+x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\log \left (\frac {x}{3}\right )}{-2+x} \, dx,x,2+x\right )\\ &=\frac {x}{2}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )+\frac {2}{3} \log \left (\frac {3}{2}\right ) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)-\frac {1}{3} (2+x) \log \left (\frac {2+x}{3}\right )+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {2 \text {Li}_2\left (-\frac {x}{2}\right )}{3}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx,x,2+x\right )\\ &=\frac {x}{2}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )+\frac {2}{3} \log \left (\frac {3}{2}\right ) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)-\frac {1}{3} (2+x) \log \left (\frac {2+x}{3}\right )+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{6} \left (3 x+2 x (18+3 x+\log (2 x)) \log \left (\frac {2+x}{3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 39*x + 6*x^2 + (76 + 62*x + 12*x^2)*Log[(2 + x)/3] + Log[2*x]*(2*x + (4 + 2*x)*Log[(2 + x)/3]))
/(12 + 6*x),x]

[Out]

(3*x + 2*x*(18 + 3*x + Log[2*x])*Log[(2 + x)/3])/6

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 31, normalized size = 1.03 \begin {gather*} \frac {1}{3} \, x \log \left (2 \, x\right ) \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + {\left (x^{2} + 6 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*log(2/3+1/3*x)+2*x)*log(2*x)+(12*x^2+62*x+76)*log(2/3+1/3*x)+6*x^2+39*x+6)/(6*x+12),x, alg
orithm="fricas")

[Out]

1/3*x*log(2*x)*log(1/3*x + 2/3) + (x^2 + 6*x)*log(1/3*x + 2/3) + 1/2*x

________________________________________________________________________________________

giac [A]  time = 0.13, size = 47, normalized size = 1.57 \begin {gather*} -x^{2} \log \relax (3) - \frac {1}{2} \, x {\left (12 \, \log \relax (3) - 1\right )} - \frac {1}{3} \, {\left (x \log \relax (3) - x \log \left (x + 2\right )\right )} \log \left (2 \, x\right ) + {\left (x^{2} + 6 \, x\right )} \log \left (x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*log(2/3+1/3*x)+2*x)*log(2*x)+(12*x^2+62*x+76)*log(2/3+1/3*x)+6*x^2+39*x+6)/(6*x+12),x, alg
orithm="giac")

[Out]

-x^2*log(3) - 1/2*x*(12*log(3) - 1) - 1/3*(x*log(3) - x*log(x + 2))*log(2*x) + (x^2 + 6*x)*log(x + 2)

________________________________________________________________________________________

maple [A]  time = 0.40, size = 26, normalized size = 0.87

method	result	size
risch	\(\left (\frac {x \ln \left (2 x \right )}{3}+x^{2}+6 x \right ) \ln \left (\frac {2}{3}+\frac {x}{3}\right )+\frac {x}{2}\)	\(26\)
norman	\(\ln \left (\frac {2}{3}+\frac {x}{3}\right ) x^{2}+\frac {x}{2}+6 \ln \left (\frac {2}{3}+\frac {x}{3}\right ) x +\frac {\ln \left (2 x \right ) \ln \left (\frac {2}{3}+\frac {x}{3}\right ) x}{3}\)	\(37\)
default	\(\left (2+x \right )^{2} \ln \left (2+x \right )+\frac {x}{2}-\frac {20}{3}+\frac {7 \left (2+x \right ) \ln \left (2+x \right )}{3}-\frac {26 \ln \left (2+x \right )}{3}-\frac {x \ln \left (2+x \right )}{3}+\frac {x \ln \relax (x ) \ln \left (2+x \right )}{3}+\frac {x \ln \relax (2) \ln \left (2+x \right )}{3}-\frac {x \ln \relax (2) \ln \relax (3)}{3}-\frac {2 \ln \relax (2)}{3}-\frac {x \ln \relax (3) \ln \relax (x )}{3}-x^{2} \ln \relax (3)-6 x \ln \relax (3)\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x+4)*ln(2/3+1/3*x)+2*x)*ln(2*x)+(12*x^2+62*x+76)*ln(2/3+1/3*x)+6*x^2+39*x+6)/(6*x+12),x,method=_RETUR
NVERBOSE)

[Out]

(1/3*x*ln(2*x)+x^2+6*x)*ln(2/3+1/3*x)+1/2*x

________________________________________________________________________________________

maxima [B]  time = 0.52, size = 101, normalized size = 3.37 \begin {gather*} -\frac {1}{3} \, x \log \relax (3) \log \relax (x) - \frac {1}{3} \, {\left (\log \relax (3) \log \relax (2) - \log \relax (3) - 1\right )} x + \frac {1}{3} \, {\left (x {\left (\log \relax (2) - 1\right )} + x \log \relax (x) - 2\right )} \log \left (x + 2\right ) - \frac {38}{3} \, \log \relax (3) \log \left (x + 2\right ) + \frac {38}{3} \, \log \left (x + 2\right )^{2} + {\left (x^{2} - 4 \, x + 8 \, \log \left (x + 2\right )\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {31}{3} \, {\left (x - 2 \, \log \left (x + 2\right )\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{6} \, x + \frac {2}{3} \, \log \left (x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*log(2/3+1/3*x)+2*x)*log(2*x)+(12*x^2+62*x+76)*log(2/3+1/3*x)+6*x^2+39*x+6)/(6*x+12),x, alg
orithm="maxima")

[Out]

-1/3*x*log(3)*log(x) - 1/3*(log(3)*log(2) - log(3) - 1)*x + 1/3*(x*(log(2) - 1) + x*log(x) - 2)*log(x + 2) - 3
8/3*log(3)*log(x + 2) + 38/3*log(x + 2)^2 + (x^2 - 4*x + 8*log(x + 2))*log(1/3*x + 2/3) + 31/3*(x - 2*log(x +
2))*log(1/3*x + 2/3) + 1/6*x + 2/3*log(x + 2)

________________________________________________________________________________________

mupad [B]  time = 4.24, size = 25, normalized size = 0.83 \begin {gather*} \frac {x}{2}+\ln \left (\frac {x}{3}+\frac {2}{3}\right )\,\left (6\,x+\frac {x\,\ln \left (2\,x\right )}{3}+x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((39*x + log(2*x)*(2*x + log(x/3 + 2/3)*(2*x + 4)) + log(x/3 + 2/3)*(62*x + 12*x^2 + 76) + 6*x^2 + 6)/(6*x
+ 12),x)

[Out]

x/2 + log(x/3 + 2/3)*(6*x + (x*log(2*x))/3 + x^2)

________________________________________________________________________________________

sympy [A]  time = 0.85, size = 37, normalized size = 1.23 \begin {gather*} \frac {x}{2} + \left (x^{2} + \frac {x \log {\left (2 x \right )}}{3} + 6 x + \frac {9}{2}\right ) \log {\left (\frac {x}{3} + \frac {2}{3} \right )} - \frac {9 \log {\left (x + 2 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+4)*ln(2/3+1/3*x)+2*x)*ln(2*x)+(12*x**2+62*x+76)*ln(2/3+1/3*x)+6*x**2+39*x+6)/(6*x+12),x)

[Out]

x/2 + (x**2 + x*log(2*x)/3 + 6*x + 9/2)*log(x/3 + 2/3) - 9*log(x + 2)/2

________________________________________________________________________________________