Integrand size = 14, antiderivative size = 606 \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=-\frac {302177 x}{1152}+\frac {8029 x^2}{2304}-\frac {763 x^3}{3456}+\frac {3 x^4}{256}+\frac {377}{64} (2+x)^2-\frac {71}{216} (2+x)^3+\frac {3}{256} (2+x)^4+\frac {2069}{144} \log (2+x)-\frac {187}{64} x^2 \log (2+x)+\frac {83}{288} x^3 \log (2+x)-\frac {3}{128} x^4 \log (2+x)+\frac {6733}{32} (2+x) \log (2+x)-\frac {377}{32} (2+x)^2 \log (2+x)+\frac {71}{72} (2+x)^3 \log (2+x)-\frac {3}{64} (2+x)^4 \log (2+x)-\frac {43}{12} \log ^2(2+x)-\frac {17}{48} x^3 \log ^2(2+x)+\frac {3}{64} x^4 \log ^2(2+x)-\frac {1251}{16} (2+x) \log ^2(2+x)+\frac {273}{32} (2+x)^2 \log ^2(2+x)-\frac {3}{4} (2+x)^3 \log ^2(2+x)+\frac {3}{64} (2+x)^4 \log ^2(2+x)+\frac {65}{4} (2+x) \log ^3(2+x)-\frac {33}{8} (2+x)^2 \log ^3(2+x)+\frac {3}{4} (2+x)^3 \log ^3(2+x)-\frac {1}{16} (2+x)^4 \log ^3(2+x)+\frac {3891}{128} \log (3+x)-\frac {115}{48} x^2 \log (3+x)+\frac {37}{144} x^3 \log (3+x)-\frac {3}{128} x^4 \log (3+x)+\frac {415}{12} (3+x) \log (3+x)-\frac {4083}{32} \log (2+x) \log (3+x)-25 x \log (2+x) \log (3+x)+\frac {13}{4} x^2 \log (2+x) \log (3+x)-\frac {7}{12} x^3 \log (2+x) \log (3+x)+\frac {3}{32} x^4 \log (2+x) \log (3+x)+\frac {963}{16} \log ^2(2+x) \log (3+x)+6 x \log ^2(2+x) \log (3+x)-\frac {3}{2} x^2 \log ^2(2+x) \log (3+x)+\frac {1}{2} x^3 \log ^2(2+x) \log (3+x)-\frac {3}{16} x^4 \log ^2(2+x) \log (3+x)-\frac {81}{4} \log ^3(2+x) \log (3+x)+\frac {1}{4} x^4 \log ^3(2+x) \log (3+x)-\frac {5609 \operatorname {PolyLog}(2,-2-x)}{96}+\frac {563}{8} \log (2+x) \operatorname {PolyLog}(2,-2-x)-\frac {195}{4} \log ^2(2+x) \operatorname {PolyLog}(2,-2-x)-\frac {563 \operatorname {PolyLog}(3,-2-x)}{8}+\frac {195}{2} \log (2+x) \operatorname {PolyLog}(3,-2-x)-\frac {195 \operatorname {PolyLog}(4,-2-x)}{2} \]
-302177/1152*x-43/12*ln(2+x)^2-5609/96*polylog(2,-2-x)-563/8*polylog(3,-2- x)-195/2*polylog(4,-2-x)+377/64*(2+x)^2-71/216*(2+x)^3+3/256*(2+x)^4+2069/ 144*ln(2+x)+3891/128*ln(3+x)+3/256*x^4+8029/2304*x^2-763/3456*x^3+963/16*l n(2+x)^2*ln(3+x)-81/4*ln(2+x)^3*ln(3+x)+563/8*ln(2+x)*polylog(2,-2-x)-195/ 4*ln(2+x)^2*polylog(2,-2-x)+195/2*ln(2+x)*polylog(3,-2-x)-187/64*x^2*ln(2+ x)+83/288*x^3*ln(2+x)-3/128*x^4*ln(2+x)+6733/32*(2+x)*ln(2+x)-377/32*(2+x) ^2*ln(2+x)+71/72*(2+x)^3*ln(2+x)-3/64*(2+x)^4*ln(2+x)-17/48*x^3*ln(2+x)^2+ 3/64*x^4*ln(2+x)^2-1251/16*(2+x)*ln(2+x)^2+273/32*(2+x)^2*ln(2+x)^2-3/4*(2 +x)^3*ln(2+x)^2+3/64*(2+x)^4*ln(2+x)^2+65/4*(2+x)*ln(2+x)^3-33/8*(2+x)^2*l n(2+x)^3+3/4*(2+x)^3*ln(2+x)^3-1/16*(2+x)^4*ln(2+x)^3-115/48*x^2*ln(3+x)+3 7/144*x^3*ln(3+x)-3/128*x^4*ln(3+x)+415/12*(3+x)*ln(3+x)-4083/32*ln(2+x)*l n(3+x)+1/2*x^3*ln(2+x)^2*ln(3+x)+13/4*x^2*ln(2+x)*ln(3+x)-25*x*ln(2+x)*ln( 3+x)-3/2*x^2*ln(2+x)^2*ln(3+x)+6*x*ln(2+x)^2*ln(3+x)-7/12*x^3*ln(2+x)*ln(3 +x)+3/32*x^4*ln(2+x)*ln(3+x)-3/16*x^4*ln(2+x)^2*ln(3+x)+1/4*x^4*ln(2+x)^3* ln(3+x)
Time = 0.30 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.68 \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\frac {-195984-558290 x+17705 x^2-1050 x^3+54 x^4+910528 \log (2+x)+400008 x \log (2+x)-22836 x^2 \log (2+x)+2072 x^3 \log (2+x)-162 x^4 \log (2+x)-302016 \log ^2(2+x)-118800 x \log ^2(2+x)+11880 x^2 \log ^2(2+x)-1680 x^3 \log ^2(2+x)+216 x^4 \log ^2(2+x)+48384 \log ^3(2+x)+15552 x \log ^3(2+x)-2592 x^2 \log ^3(2+x)+576 x^3 \log ^3(2+x)-144 x^4 \log ^3(2+x)+309078 \log (3+x)+79680 x \log (3+x)-5520 x^2 \log (3+x)+592 x^3 \log (3+x)-54 x^4 \log (3+x)-293976 \log (2+x) \log (3+x)-57600 x \log (2+x) \log (3+x)+7488 x^2 \log (2+x) \log (3+x)-1344 x^3 \log (2+x) \log (3+x)+216 x^4 \log (2+x) \log (3+x)+138672 \log ^2(2+x) \log (3+x)+13824 x \log ^2(2+x) \log (3+x)-3456 x^2 \log ^2(2+x) \log (3+x)+1152 x^3 \log ^2(2+x) \log (3+x)-432 x^4 \log ^2(2+x) \log (3+x)-46656 \log ^3(2+x) \log (3+x)+576 x^4 \log ^3(2+x) \log (3+x)-24 \left (5609-6756 \log (2+x)+4680 \log ^2(2+x)\right ) \operatorname {PolyLog}(2,-2-x)+288 (-563+780 \log (2+x)) \operatorname {PolyLog}(3,-2-x)-224640 \operatorname {PolyLog}(4,-2-x)}{2304} \]
(-195984 - 558290*x + 17705*x^2 - 1050*x^3 + 54*x^4 + 910528*Log[2 + x] + 400008*x*Log[2 + x] - 22836*x^2*Log[2 + x] + 2072*x^3*Log[2 + x] - 162*x^4 *Log[2 + x] - 302016*Log[2 + x]^2 - 118800*x*Log[2 + x]^2 + 11880*x^2*Log[ 2 + x]^2 - 1680*x^3*Log[2 + x]^2 + 216*x^4*Log[2 + x]^2 + 48384*Log[2 + x] ^3 + 15552*x*Log[2 + x]^3 - 2592*x^2*Log[2 + x]^3 + 576*x^3*Log[2 + x]^3 - 144*x^4*Log[2 + x]^3 + 309078*Log[3 + x] + 79680*x*Log[3 + x] - 5520*x^2* Log[3 + x] + 592*x^3*Log[3 + x] - 54*x^4*Log[3 + x] - 293976*Log[2 + x]*Lo g[3 + x] - 57600*x*Log[2 + x]*Log[3 + x] + 7488*x^2*Log[2 + x]*Log[3 + x] - 1344*x^3*Log[2 + x]*Log[3 + x] + 216*x^4*Log[2 + x]*Log[3 + x] + 138672* Log[2 + x]^2*Log[3 + x] + 13824*x*Log[2 + x]^2*Log[3 + x] - 3456*x^2*Log[2 + x]^2*Log[3 + x] + 1152*x^3*Log[2 + x]^2*Log[3 + x] - 432*x^4*Log[2 + x] ^2*Log[3 + x] - 46656*Log[2 + x]^3*Log[3 + x] + 576*x^4*Log[2 + x]^3*Log[3 + x] - 24*(5609 - 6756*Log[2 + x] + 4680*Log[2 + x]^2)*PolyLog[2, -2 - x] + 288*(-563 + 780*Log[2 + x])*PolyLog[3, -2 - x] - 224640*PolyLog[4, -2 - x])/2304
Time = 4.52 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2889, 2863, 2009, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \log ^3(x+2) \log (x+3) \, dx\) |
| \(\Big \downarrow \) 2889 |
| \(\displaystyle -\frac {1}{4} \int \frac {x^4 \log ^3(x+2)}{x+3}dx-\frac {3}{4} \int \frac {x^4 \log ^2(x+2) \log (x+3)}{x+2}dx+\frac {1}{4} x^4 \log ^3(x+2) \log (x+3)\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle -\frac {3}{4} \int \frac {x^4 \log ^2(x+2) \log (x+3)}{x+2}dx-\frac {1}{4} \int \left (x^3 \log ^3(x+2)-3 x^2 \log ^3(x+2)+9 x \log ^3(x+2)+\frac {81 \log ^3(x+2)}{x+3}-27 \log ^3(x+2)\right )dx+\frac {1}{4} x^4 \log ^3(x+2) \log (x+3)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{4} \int \frac {x^4 \log ^2(x+2) \log (x+3)}{x+2}dx+\frac {1}{4} \left (-486 \operatorname {PolyLog}(4,-x-2)-243 \operatorname {PolyLog}(2,-x-2) \log ^2(x+2)+486 \operatorname {PolyLog}(3,-x-2) \log (x+2)+\frac {3}{128} (x+2)^4-\frac {2}{3} (x+2)^3+\frac {99}{8} (x+2)^2-390 x-\frac {1}{4} (x+2)^4 \log ^3(x+2)+3 (x+2)^3 \log ^3(x+2)-\frac {33}{2} (x+2)^2 \log ^3(x+2)+65 (x+2) \log ^3(x+2)-81 \log ^3(x+2) \log (x+3)+\frac {3}{16} (x+2)^4 \log ^2(x+2)-3 (x+2)^3 \log ^2(x+2)+\frac {99}{4} (x+2)^2 \log ^2(x+2)-195 (x+2) \log ^2(x+2)-\frac {3}{32} (x+2)^4 \log (x+2)+2 (x+2)^3 \log (x+2)-\frac {99}{4} (x+2)^2 \log (x+2)+390 (x+2) \log (x+2)\right )+\frac {1}{4} x^4 \log ^3(x+2) \log (x+3)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{4} \int \left (\log ^2(x+2) \log (x+3) x^3-2 \log ^2(x+2) \log (x+3) x^2+4 \log ^2(x+2) \log (x+3) x+\frac {16 \log ^2(x+2) \log (x+3)}{x+2}-8 \log ^2(x+2) \log (x+3)\right )dx+\frac {1}{4} \left (-486 \operatorname {PolyLog}(4,-x-2)-243 \operatorname {PolyLog}(2,-x-2) \log ^2(x+2)+486 \operatorname {PolyLog}(3,-x-2) \log (x+2)+\frac {3}{128} (x+2)^4-\frac {2}{3} (x+2)^3+\frac {99}{8} (x+2)^2-390 x-\frac {1}{4} (x+2)^4 \log ^3(x+2)+3 (x+2)^3 \log ^3(x+2)-\frac {33}{2} (x+2)^2 \log ^3(x+2)+65 (x+2) \log ^3(x+2)-81 \log ^3(x+2) \log (x+3)+\frac {3}{16} (x+2)^4 \log ^2(x+2)-3 (x+2)^3 \log ^2(x+2)+\frac {99}{4} (x+2)^2 \log ^2(x+2)-195 (x+2) \log ^2(x+2)-\frac {3}{32} (x+2)^4 \log (x+2)+2 (x+2)^3 \log (x+2)-\frac {99}{4} (x+2)^2 \log (x+2)+390 (x+2) \log (x+2)\right )+\frac {1}{4} x^4 \log ^3(x+2) \log (x+3)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{4} \left (\frac {5609 \operatorname {PolyLog}(2,-x-2)}{72}+\frac {563 \operatorname {PolyLog}(3,-x-2)}{6}-32 \operatorname {PolyLog}(4,-x-2)-16 \operatorname {PolyLog}(2,-x-2) \log ^2(x+2)-\frac {563}{6} \operatorname {PolyLog}(2,-x-2) \log (x+2)+32 \operatorname {PolyLog}(3,-x-2) \log (x+2)-\frac {x^4}{64}-\frac {1}{16} x^4 \log ^2(x+2)+\frac {1}{4} x^4 \log ^2(x+2) \log (x+3)+\frac {1}{32} x^4 \log (x+2)-\frac {1}{8} x^4 \log (x+2) \log (x+3)+\frac {1}{32} x^4 \log (x+3)+\frac {763 x^3}{2592}+\frac {17}{36} x^3 \log ^2(x+2)-\frac {2}{3} x^3 \log ^2(x+2) \log (x+3)-\frac {83}{216} x^3 \log (x+2)+\frac {7}{9} x^3 \log (x+2) \log (x+3)-\frac {37}{108} x^3 \log (x+3)-\frac {8029 x^2}{1728}+2 x^2 \log ^2(x+2) \log (x+3)+\frac {187}{48} x^2 \log (x+2)-\frac {13}{3} x^2 \log (x+2) \log (x+3)+\frac {115}{36} x^2 \log (x+3)+\frac {189857 x}{864}-\frac {1}{128} (x+2)^4+\frac {35}{162} (x+2)^3-\frac {179}{48} (x+2)^2-8 x \log ^2(x+2) \log (x+3)-\frac {25}{8} (x+2)^2 \log ^2(x+2)+\frac {157}{4} (x+2) \log ^2(x+2)+\frac {43}{9} \log ^2(x+2)-\frac {321}{4} \log ^2(x+2) \log (x+3)+\frac {100}{3} x \log (x+2) \log (x+3)+\frac {1}{32} (x+2)^4 \log (x+2)-\frac {35}{54} (x+2)^3 \log (x+2)+\frac {179}{24} (x+2)^2 \log (x+2)-\frac {3613}{24} (x+2) \log (x+2)-\frac {2069}{108} \log (x+2)-\frac {415}{9} (x+3) \log (x+3)+\frac {1361}{8} \log (x+2) \log (x+3)-\frac {1297}{32} \log (x+3)\right )+\frac {1}{4} \left (-486 \operatorname {PolyLog}(4,-x-2)-243 \operatorname {PolyLog}(2,-x-2) \log ^2(x+2)+486 \operatorname {PolyLog}(3,-x-2) \log (x+2)+\frac {3}{128} (x+2)^4-\frac {2}{3} (x+2)^3+\frac {99}{8} (x+2)^2-390 x-\frac {1}{4} (x+2)^4 \log ^3(x+2)+3 (x+2)^3 \log ^3(x+2)-\frac {33}{2} (x+2)^2 \log ^3(x+2)+65 (x+2) \log ^3(x+2)-81 \log ^3(x+2) \log (x+3)+\frac {3}{16} (x+2)^4 \log ^2(x+2)-3 (x+2)^3 \log ^2(x+2)+\frac {99}{4} (x+2)^2 \log ^2(x+2)-195 (x+2) \log ^2(x+2)-\frac {3}{32} (x+2)^4 \log (x+2)+2 (x+2)^3 \log (x+2)-\frac {99}{4} (x+2)^2 \log (x+2)+390 (x+2) \log (x+2)\right )+\frac {1}{4} x^4 \log ^3(x+2) \log (x+3)\) |
(x^4*Log[2 + x]^3*Log[3 + x])/4 + (-390*x + (99*(2 + x)^2)/8 - (2*(2 + x)^ 3)/3 + (3*(2 + x)^4)/128 + 390*(2 + x)*Log[2 + x] - (99*(2 + x)^2*Log[2 + x])/4 + 2*(2 + x)^3*Log[2 + x] - (3*(2 + x)^4*Log[2 + x])/32 - 195*(2 + x) *Log[2 + x]^2 + (99*(2 + x)^2*Log[2 + x]^2)/4 - 3*(2 + x)^3*Log[2 + x]^2 + (3*(2 + x)^4*Log[2 + x]^2)/16 + 65*(2 + x)*Log[2 + x]^3 - (33*(2 + x)^2*L og[2 + x]^3)/2 + 3*(2 + x)^3*Log[2 + x]^3 - ((2 + x)^4*Log[2 + x]^3)/4 - 8 1*Log[2 + x]^3*Log[3 + x] - 243*Log[2 + x]^2*PolyLog[2, -2 - x] + 486*Log[ 2 + x]*PolyLog[3, -2 - x] - 486*PolyLog[4, -2 - x])/4 - (3*((189857*x)/864 - (8029*x^2)/1728 + (763*x^3)/2592 - x^4/64 - (179*(2 + x)^2)/48 + (35*(2 + x)^3)/162 - (2 + x)^4/128 - (2069*Log[2 + x])/108 + (187*x^2*Log[2 + x] )/48 - (83*x^3*Log[2 + x])/216 + (x^4*Log[2 + x])/32 - (3613*(2 + x)*Log[2 + x])/24 + (179*(2 + x)^2*Log[2 + x])/24 - (35*(2 + x)^3*Log[2 + x])/54 + ((2 + x)^4*Log[2 + x])/32 + (43*Log[2 + x]^2)/9 + (17*x^3*Log[2 + x]^2)/3 6 - (x^4*Log[2 + x]^2)/16 + (157*(2 + x)*Log[2 + x]^2)/4 - (25*(2 + x)^2*L og[2 + x]^2)/8 - (1297*Log[3 + x])/32 + (115*x^2*Log[3 + x])/36 - (37*x^3* Log[3 + x])/108 + (x^4*Log[3 + x])/32 - (415*(3 + x)*Log[3 + x])/9 + (1361 *Log[2 + x]*Log[3 + x])/8 + (100*x*Log[2 + x]*Log[3 + x])/3 - (13*x^2*Log[ 2 + x]*Log[3 + x])/3 + (7*x^3*Log[2 + x]*Log[3 + x])/9 - (x^4*Log[2 + x]*L og[3 + x])/8 - (321*Log[2 + x]^2*Log[3 + x])/4 - 8*x*Log[2 + x]^2*Log[3 + x] + 2*x^2*Log[2 + x]^2*Log[3 + x] - (2*x^3*Log[2 + x]^2*Log[3 + x])/3 ...
3.1.29.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^( r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1)), x ] + (-Simp[g*j*(m/(r + 1)) Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Simp[b*e*n*(p/(r + 1)) Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*((f + g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /; FreeQ[ {a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (E qQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
\[\int x^{3} \ln \left (2+x \right )^{3} \ln \left (3+x \right )d x\]
\[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\int { x^{3} \log \left (x + 3\right ) \log \left (x + 2\right )^{3} \,d x } \]
\[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\left (\frac {x^{4} \log {\left (x + 2 \right )}^{3}}{4} - \frac {3 x^{4} \log {\left (x + 2 \right )}^{2}}{16} + \frac {3 x^{4} \log {\left (x + 2 \right )}}{32} - \frac {3 x^{4}}{128} + \frac {x^{3} \log {\left (x + 2 \right )}^{2}}{2} - \frac {7 x^{3} \log {\left (x + 2 \right )}}{12} + \frac {37 x^{3}}{144} - \frac {3 x^{2} \log {\left (x + 2 \right )}^{2}}{2} + \frac {13 x^{2} \log {\left (x + 2 \right )}}{4} - \frac {115 x^{2}}{48} + 6 x \log {\left (x + 2 \right )}^{2} - 25 x \log {\left (x + 2 \right )} + \frac {415 x}{12} - 4 \log {\left (x + 2 \right )}^{3} + 25 \log {\left (x + 2 \right )}^{2} - \frac {415 \log {\left (x + 2 \right )}}{6} + \frac {10955281}{240000}\right ) \log {\left (x + 3 \right )} - \frac {\int \frac {24900000 x}{x + 3}\, dx + \int \left (- \frac {1725000 x^{2}}{x + 3}\right )\, dx + \int \frac {185000 x^{3}}{x + 3}\, dx + \int \left (- \frac {16875 x^{4}}{x + 3}\right )\, dx + \int \left (- \frac {49800000 \log {\left (x + 2 \right )}}{x + 3}\right )\, dx + \int \frac {18000000 \log {\left (x + 2 \right )}^{2}}{x + 3}\, dx + \int \left (- \frac {2880000 \log {\left (x + 2 \right )}^{3}}{x + 3}\right )\, dx + \int \left (- \frac {18000000 x \log {\left (x + 2 \right )}}{x + 3}\right )\, dx + \int \frac {4320000 x \log {\left (x + 2 \right )}^{2}}{x + 3}\, dx + \int \frac {2340000 x^{2} \log {\left (x + 2 \right )}}{x + 3}\, dx + \int \left (- \frac {1080000 x^{2} \log {\left (x + 2 \right )}^{2}}{x + 3}\right )\, dx + \int \left (- \frac {420000 x^{3} \log {\left (x + 2 \right )}}{x + 3}\right )\, dx + \int \frac {360000 x^{3} \log {\left (x + 2 \right )}^{2}}{x + 3}\, dx + \int \frac {67500 x^{4} \log {\left (x + 2 \right )}}{x + 3}\, dx + \int \left (- \frac {135000 x^{4} \log {\left (x + 2 \right )}^{2}}{x + 3}\right )\, dx + \int \frac {180000 x^{4} \log {\left (x + 2 \right )}^{3}}{x + 3}\, dx + \int \frac {32865843}{x + 3}\, dx}{720000} \]
(x**4*log(x + 2)**3/4 - 3*x**4*log(x + 2)**2/16 + 3*x**4*log(x + 2)/32 - 3 *x**4/128 + x**3*log(x + 2)**2/2 - 7*x**3*log(x + 2)/12 + 37*x**3/144 - 3* x**2*log(x + 2)**2/2 + 13*x**2*log(x + 2)/4 - 115*x**2/48 + 6*x*log(x + 2) **2 - 25*x*log(x + 2) + 415*x/12 - 4*log(x + 2)**3 + 25*log(x + 2)**2 - 41 5*log(x + 2)/6 + 10955281/240000)*log(x + 3) - (Integral(24900000*x/(x + 3 ), x) + Integral(-1725000*x**2/(x + 3), x) + Integral(185000*x**3/(x + 3), x) + Integral(-16875*x**4/(x + 3), x) + Integral(-49800000*log(x + 2)/(x + 3), x) + Integral(18000000*log(x + 2)**2/(x + 3), x) + Integral(-2880000 *log(x + 2)**3/(x + 3), x) + Integral(-18000000*x*log(x + 2)/(x + 3), x) + Integral(4320000*x*log(x + 2)**2/(x + 3), x) + Integral(2340000*x**2*log( x + 2)/(x + 3), x) + Integral(-1080000*x**2*log(x + 2)**2/(x + 3), x) + In tegral(-420000*x**3*log(x + 2)/(x + 3), x) + Integral(360000*x**3*log(x + 2)**2/(x + 3), x) + Integral(67500*x**4*log(x + 2)/(x + 3), x) + Integral( -135000*x**4*log(x + 2)**2/(x + 3), x) + Integral(180000*x**4*log(x + 2)** 3/(x + 3), x) + Integral(32865843/(x + 3), x))/720000
Time = 0.21 (sec) , antiderivative size = 518, normalized size of antiderivative = 0.85 \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\frac {3}{128} \, x^{4} + \frac {1}{16} \, {\left (4 \, x^{4} \log \left (x + 3\right ) - x^{4} + 4 \, x^{3} - 18 \, x^{2} + 108 \, x - 324 \, \log \left (x + 3\right )\right )} \log \left (x + 2\right )^{3} - \frac {65}{4} \, \log \left (x + 3\right ) \log \left (x + 2\right )^{3} + \frac {195}{4} \, \log \left (x + 3\right ) \log \left (x + 2\right )^{2} \log \left (-x - 2\right ) - \frac {175}{384} \, x^{3} + \frac {1}{96} \, {\left (9 \, x^{4} - 70 \, x^{3} + 495 \, x^{2} - 6 \, {\left (3 \, x^{4} - 8 \, x^{3} + 24 \, x^{2} - 96 \, x\right )} \log \left (x + 3\right ) + 4680 \, \log \left (x + 3\right ) \log \left (-x - 2\right ) - 4950 \, x + 4680 \, {\rm Li}_2\left (x + 3\right ) + 5778 \, \log \left (x + 3\right ) + 6048 \, \log \left (x + 2\right )\right )} \log \left (x + 2\right )^{2} + \frac {195}{4} \, {\rm Li}_2\left (x + 3\right ) \log \left (x + 2\right )^{2} - \frac {195}{4} \, {\rm Li}_2\left (-x - 2\right ) \log \left (x + 2\right )^{2} + \frac {563}{16} \, \log \left (x + 3\right ) \log \left (x + 2\right )^{2} + 21 \, \log \left (x + 2\right )^{3} + \frac {17705}{2304} \, x^{2} + \frac {1}{8} \, {\left (780 \, \log \left (x + 2\right )^{2} - 563 \, \log \left (x + 2\right )\right )} {\rm Li}_2\left (-x - 2\right ) - \frac {1}{1152} \, {\left (27 \, x^{4} - 296 \, x^{3} - 18720 \, \log \left (x + 2\right )^{3} + 2760 \, x^{2} + 40536 \, \log \left (x + 2\right )^{2} - 39840 \, x - 67308 \, \log \left (x + 2\right )\right )} \log \left (x + 3\right ) - \frac {1}{1152} \, {\left (81 \, x^{4} - 1036 \, x^{3} + 56160 \, \log \left (x + 3\right ) \log \left (x + 2\right )^{2} + 112320 \, \log \left (x + 3\right ) \log \left (x + 2\right ) \log \left (-x - 2\right ) + 11418 \, x^{2} - 12 \, {\left (9 \, x^{4} - 56 \, x^{3} + 312 \, x^{2} + 4680 \, \log \left (x + 2\right )^{2} - 2400 \, x - 6756 \, \log \left (x + 2\right )\right )} \log \left (x + 3\right ) + 112320 \, {\rm Li}_2\left (x + 3\right ) \log \left (x + 2\right ) + 112320 \, {\rm Li}_2\left (-x - 2\right ) \log \left (x + 2\right ) - 81072 \, \log \left (x + 3\right ) \log \left (x + 2\right ) + 72576 \, \log \left (x + 2\right )^{2} - 200004 \, x - 81072 \, {\rm Li}_2\left (-x - 2\right ) + 146988 \, \log \left (x + 3\right ) + 302016 \, \log \left (x + 2\right ) - 112320 \, {\rm Li}_{3}(-x - 2)\right )} \log \left (x + 2\right ) + \frac {563}{8} \, {\rm Li}_2\left (-x - 2\right ) \log \left (x + 2\right ) - \frac {5609}{96} \, \log \left (x + 3\right ) \log \left (x + 2\right ) + \frac {1573}{12} \, \log \left (x + 2\right )^{2} - \frac {279145}{1152} \, x - \frac {5609}{96} \, {\rm Li}_2\left (-x - 2\right ) + \frac {17171}{128} \, \log \left (x + 3\right ) + \frac {14227}{36} \, \log \left (x + 2\right ) - \frac {195}{2} \, {\rm Li}_{4}(-x - 2) - \frac {563}{8} \, {\rm Li}_{3}(-x - 2) \]
3/128*x^4 + 1/16*(4*x^4*log(x + 3) - x^4 + 4*x^3 - 18*x^2 + 108*x - 324*lo g(x + 3))*log(x + 2)^3 - 65/4*log(x + 3)*log(x + 2)^3 + 195/4*log(x + 3)*l og(x + 2)^2*log(-x - 2) - 175/384*x^3 + 1/96*(9*x^4 - 70*x^3 + 495*x^2 - 6 *(3*x^4 - 8*x^3 + 24*x^2 - 96*x)*log(x + 3) + 4680*log(x + 3)*log(-x - 2) - 4950*x + 4680*dilog(x + 3) + 5778*log(x + 3) + 6048*log(x + 2))*log(x + 2)^2 + 195/4*dilog(x + 3)*log(x + 2)^2 - 195/4*dilog(-x - 2)*log(x + 2)^2 + 563/16*log(x + 3)*log(x + 2)^2 + 21*log(x + 2)^3 + 17705/2304*x^2 + 1/8* (780*log(x + 2)^2 - 563*log(x + 2))*dilog(-x - 2) - 1/1152*(27*x^4 - 296*x ^3 - 18720*log(x + 2)^3 + 2760*x^2 + 40536*log(x + 2)^2 - 39840*x - 67308* log(x + 2))*log(x + 3) - 1/1152*(81*x^4 - 1036*x^3 + 56160*log(x + 3)*log( x + 2)^2 + 112320*log(x + 3)*log(x + 2)*log(-x - 2) + 11418*x^2 - 12*(9*x^ 4 - 56*x^3 + 312*x^2 + 4680*log(x + 2)^2 - 2400*x - 6756*log(x + 2))*log(x + 3) + 112320*dilog(x + 3)*log(x + 2) + 112320*dilog(-x - 2)*log(x + 2) - 81072*log(x + 3)*log(x + 2) + 72576*log(x + 2)^2 - 200004*x - 81072*dilog (-x - 2) + 146988*log(x + 3) + 302016*log(x + 2) - 112320*polylog(3, -x - 2))*log(x + 2) + 563/8*dilog(-x - 2)*log(x + 2) - 5609/96*log(x + 3)*log(x + 2) + 1573/12*log(x + 2)^2 - 279145/1152*x - 5609/96*dilog(-x - 2) + 171 71/128*log(x + 3) + 14227/36*log(x + 2) - 195/2*polylog(4, -x - 2) - 563/8 *polylog(3, -x - 2)
\[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\int { x^{3} \log \left (x + 3\right ) \log \left (x + 2\right )^{3} \,d x } \]
Timed out. \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\int x^3\,{\ln \left (x+2\right )}^3\,\ln \left (x+3\right ) \,d x \]