Integrand size = 14, antiderivative size = 606 \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=-\frac {302177 x}{1152}+\frac {8029 x^2}{2304}-\frac {763 x^3}{3456}+\frac {3 x^4}{256}+\frac {377}{64} (2+x)^2-\frac {71}{216} (2+x)^3+\frac {3}{256} (2+x)^4+\frac {2069}{144} \log (2+x)-\frac {187}{64} x^2 \log (2+x)+\frac {83}{288} x^3 \log (2+x)-\frac {3}{128} x^4 \log (2+x)+\frac {6733}{32} (2+x) \log (2+x)-\frac {377}{32} (2+x)^2 \log (2+x)+\frac {71}{72} (2+x)^3 \log (2+x)-\frac {3}{64} (2+x)^4 \log (2+x)-\frac {43}{12} \log ^2(2+x)-\frac {17}{48} x^3 \log ^2(2+x)+\frac {3}{64} x^4 \log ^2(2+x)-\frac {1251}{16} (2+x) \log ^2(2+x)+\frac {273}{32} (2+x)^2 \log ^2(2+x)-\frac {3}{4} (2+x)^3 \log ^2(2+x)+\frac {3}{64} (2+x)^4 \log ^2(2+x)+\frac {65}{4} (2+x) \log ^3(2+x)-\frac {33}{8} (2+x)^2 \log ^3(2+x)+\frac {3}{4} (2+x)^3 \log ^3(2+x)-\frac {1}{16} (2+x)^4 \log ^3(2+x)+\frac {3891}{128} \log (3+x)-\frac {115}{48} x^2 \log (3+x)+\frac {37}{144} x^3 \log (3+x)-\frac {3}{128} x^4 \log (3+x)+\frac {415}{12} (3+x) \log (3+x)-\frac {4083}{32} \log (2+x) \log (3+x)-25 x \log (2+x) \log (3+x)+\frac {13}{4} x^2 \log (2+x) \log (3+x)-\frac {7}{12} x^3 \log (2+x) \log (3+x)+\frac {3}{32} x^4 \log (2+x) \log (3+x)+\frac {963}{16} \log ^2(2+x) \log (3+x)+6 x \log ^2(2+x) \log (3+x)-\frac {3}{2} x^2 \log ^2(2+x) \log (3+x)+\frac {1}{2} x^3 \log ^2(2+x) \log (3+x)-\frac {3}{16} x^4 \log ^2(2+x) \log (3+x)-\frac {81}{4} \log ^3(2+x) \log (3+x)+\frac {1}{4} x^4 \log ^3(2+x) \log (3+x)-\frac {5609 \operatorname {PolyLog}(2,-2-x)}{96}+\frac {563}{8} \log (2+x) \operatorname {PolyLog}(2,-2-x)-\frac {195}{4} \log ^2(2+x) \operatorname {PolyLog}(2,-2-x)-\frac {563 \operatorname {PolyLog}(3,-2-x)}{8}+\frac {195}{2} \log (2+x) \operatorname {PolyLog}(3,-2-x)-\frac {195 \operatorname {PolyLog}(4,-2-x)}{2} \]
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Time = 3.53 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.00, number of steps used = 359, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.143, Rules used = {2489, 2463, 2436, 2333, 2332, 2448, 2437, 2342, 2341, 2443, 2481, 2421, 2430, 6724, 2458, 2388, 2339, 30, 2367, 2356, 45, 2372, 2338, 6874, 2479, 2440, 2438, 2441, 2442, 2445} \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=-\frac {5609 \operatorname {PolyLog}(2,-x-2)}{96}-\frac {563 \operatorname {PolyLog}(3,-x-2)}{8}-\frac {195 \operatorname {PolyLog}(4,-x-2)}{2}-\frac {195}{4} \operatorname {PolyLog}(2,-x-2) \log ^2(x+2)+\frac {563}{8} \operatorname {PolyLog}(2,-x-2) \log (x+2)+\frac {195}{2} \operatorname {PolyLog}(3,-x-2) \log (x+2)+\frac {3 x^4}{256}+\frac {1}{4} x^4 \log ^3(x+2) \log (x+3)+\frac {3}{64} x^4 \log ^2(x+2)-\frac {3}{16} x^4 \log ^2(x+2) \log (x+3)-\frac {3}{128} x^4 \log (x+2)+\frac {3}{32} x^4 \log (x+2) \log (x+3)-\frac {3}{128} x^4 \log (x+3)-\frac {763 x^3}{3456}-\frac {17}{48} x^3 \log ^2(x+2)+\frac {1}{2} x^3 \log ^2(x+2) \log (x+3)+\frac {83}{288} x^3 \log (x+2)-\frac {7}{12} x^3 \log (x+2) \log (x+3)+\frac {37}{144} x^3 \log (x+3)+\frac {8029 x^2}{2304}-\frac {3}{2} x^2 \log ^2(x+2) \log (x+3)-\frac {187}{64} x^2 \log (x+2)+\frac {13}{4} x^2 \log (x+2) \log (x+3)-\frac {115}{48} x^2 \log (x+3)-\frac {302177 x}{1152}+\frac {3}{256} (x+2)^4-\frac {71}{216} (x+2)^3+\frac {377}{64} (x+2)^2-\frac {1}{16} (x+2)^4 \log ^3(x+2)+\frac {3}{4} (x+2)^3 \log ^3(x+2)-\frac {33}{8} (x+2)^2 \log ^3(x+2)+\frac {65}{4} (x+2) \log ^3(x+2)-\frac {81}{4} \log ^3(x+2) \log (x+3)+6 x \log ^2(x+2) \log (x+3)+\frac {3}{64} (x+2)^4 \log ^2(x+2)-\frac {3}{4} (x+2)^3 \log ^2(x+2)+\frac {273}{32} (x+2)^2 \log ^2(x+2)-\frac {1251}{16} (x+2) \log ^2(x+2)-\frac {43}{12} \log ^2(x+2)+\frac {963}{16} \log ^2(x+2) \log (x+3)-25 x \log (x+2) \log (x+3)-\frac {3}{64} (x+2)^4 \log (x+2)+\frac {71}{72} (x+2)^3 \log (x+2)-\frac {377}{32} (x+2)^2 \log (x+2)+\frac {6733}{32} (x+2) \log (x+2)+\frac {2069}{144} \log (x+2)+\frac {415}{12} (x+3) \log (x+3)-\frac {4083}{32} \log (x+2) \log (x+3)+\frac {3891}{128} \log (x+3) \]
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Rule 30
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2339
Rule 2341
Rule 2342
Rule 2356
Rule 2367
Rule 2372
Rule 2388
Rule 2421
Rule 2430
Rule 2436
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2443
Rule 2445
Rule 2448
Rule 2458
Rule 2463
Rule 2479
Rule 2481
Rule 2489
Rule 6724
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \log ^3(2+x) \log (3+x)-\frac {1}{4} \int \frac {x^4 \log ^3(2+x)}{3+x} \, dx-\frac {3}{4} \int \frac {x^4 \log ^2(2+x) \log (3+x)}{2+x} \, dx \\ & = \frac {1}{4} x^4 \log ^3(2+x) \log (3+x)-\frac {1}{4} \int \left (-27 \log ^3(2+x)+9 x \log ^3(2+x)-3 x^2 \log ^3(2+x)+x^3 \log ^3(2+x)+\frac {81 \log ^3(2+x)}{3+x}\right ) \, dx-\frac {3}{4} \int \left (-8 \log ^2(2+x) \log (3+x)+4 x \log ^2(2+x) \log (3+x)-2 x^2 \log ^2(2+x) \log (3+x)+x^3 \log ^2(2+x) \log (3+x)+\frac {16 \log ^2(2+x) \log (3+x)}{2+x}\right ) \, dx \\ & = \frac {1}{4} x^4 \log ^3(2+x) \log (3+x)-\frac {1}{4} \int x^3 \log ^3(2+x) \, dx+\frac {3}{4} \int x^2 \log ^3(2+x) \, dx-\frac {3}{4} \int x^3 \log ^2(2+x) \log (3+x) \, dx+\frac {3}{2} \int x^2 \log ^2(2+x) \log (3+x) \, dx-\frac {9}{4} \int x \log ^3(2+x) \, dx-3 \int x \log ^2(2+x) \log (3+x) \, dx+6 \int \log ^2(2+x) \log (3+x) \, dx+\frac {27}{4} \int \log ^3(2+x) \, dx-12 \int \frac {\log ^2(2+x) \log (3+x)}{2+x} \, dx-\frac {81}{4} \int \frac {\log ^3(2+x)}{3+x} \, dx \\ & = 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 {3}{16} \int \frac {x^4 \log ^2(2+x)}{3+x} \, dx-\frac {1}{4} \int \left (-8 \log ^3(2+x)+12 (2+x) \log ^3(2+x)-6 (2+x)^2 \log ^3(2+x)+(2+x)^3 \log ^3(2+x)\right ) \, dx+\frac {3}{8} \int \frac {x^4 \log (2+x) \log (3+x)}{2+x} \, dx-\frac {1}{2} \int \frac {x^3 \log ^2(2+x)}{3+x} \, dx+\frac {3}{4} \int \left (4 \log ^3(2+x)-4 (2+x) \log ^3(2+x)+(2+x)^2 \log ^3(2+x)\right ) \, dx+\frac {3}{2} \int \frac {x^2 \log ^2(2+x)}{3+x} \, dx-\frac {9}{4} \int \left (-2 \log ^3(2+x)+(2+x) \log ^3(2+x)\right ) \, dx+3 \int \frac {x^2 \log (2+x) \log (3+x)}{2+x} \, dx-6 \int \frac {x \log ^2(2+x)}{3+x} \, dx+\frac {27}{4} \text {Subst}\left (\int \log ^3(x) \, dx,x,2+x\right )-12 \int \frac {x \log (2+x) \log (3+x)}{2+x} \, dx-12 \text {Subst}\left (\int \frac {\log ^2(x) \log (1+x)}{x} \, dx,x,2+x\right )+\frac {243}{4} \int \frac {\log ^2(2+x) \log (3+x)}{2+x} \, dx-\int \frac {x^3 \log (2+x) \log (3+x)}{2+x} \, dx \\ & = \frac {27}{4} (2+x) \log ^3(2+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)+12 \log ^2(2+x) \operatorname {PolyLog}(2,-2-x)+\frac {3}{16} \int \left (-27 \log ^2(2+x)+9 x \log ^2(2+x)-3 x^2 \log ^2(2+x)+x^3 \log ^2(2+x)+\frac {81 \log ^2(2+x)}{3+x}\right ) \, dx-\frac {1}{4} \int (2+x)^3 \log ^3(2+x) \, dx+\frac {3}{8} \int \left (-8 \log (2+x) \log (3+x)+4 x \log (2+x) \log (3+x)-2 x^2 \log (2+x) \log (3+x)+x^3 \log (2+x) \log (3+x)+\frac {16 \log (2+x) \log (3+x)}{2+x}\right ) \, dx-\frac {1}{2} \int \left (9 \log ^2(2+x)-3 x \log ^2(2+x)+x^2 \log ^2(2+x)-\frac {27 \log ^2(2+x)}{3+x}\right ) \, dx+\frac {3}{4} \int (2+x)^2 \log ^3(2+x) \, dx+\frac {3}{2} \int (2+x)^2 \log ^3(2+x) \, dx+\frac {3}{2} \int \left (-3 \log ^2(2+x)+x \log ^2(2+x)+\frac {9 \log ^2(2+x)}{3+x}\right ) \, dx+2 \int \log ^3(2+x) \, dx-\frac {9}{4} \int (2+x) \log ^3(2+x) \, dx+3 \int \log ^3(2+x) \, dx-2 \left (3 \int (2+x) \log ^3(2+x) \, dx\right )+3 \int \left (-2 \log (2+x) \log (3+x)+x \log (2+x) \log (3+x)+\frac {4 \log (2+x) \log (3+x)}{2+x}\right ) \, dx+\frac {9}{2} \int \log ^3(2+x) \, dx-6 \int \left (\log ^2(2+x)-\frac {3 \log ^2(2+x)}{3+x}\right ) \, dx-12 \int \left (\log (2+x) \log (3+x)-\frac {2 \log (2+x) \log (3+x)}{2+x}\right ) \, dx-\frac {81}{4} \text {Subst}\left (\int \log ^2(x) \, dx,x,2+x\right )-24 \text {Subst}\left (\int \frac {\log (x) \operatorname {PolyLog}(2,-x)}{x} \, dx,x,2+x\right )+\frac {243}{4} \text {Subst}\left (\int \frac {\log ^2(x) \log (1+x)}{x} \, dx,x,2+x\right )-\int \left (4 \log (2+x) \log (3+x)-2 x \log (2+x) \log (3+x)+x^2 \log (2+x) \log (3+x)-\frac {8 \log (2+x) \log (3+x)}{2+x}\right ) \, dx \\ & = -\frac {81}{4} (2+x) \log ^2(2+x)+\frac {27}{4} (2+x) \log ^3(2+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 {195}{4} \log ^2(2+x) \operatorname {PolyLog}(2,-2-x)-24 \log (2+x) \operatorname {PolyLog}(3,-2-x)+\frac {3}{16} \int x^3 \log ^2(2+x) \, dx-\frac {1}{4} \text {Subst}\left (\int x^3 \log ^3(x) \, dx,x,2+x\right )+\frac {3}{8} \int x^3 \log (2+x) \log (3+x) \, dx-\frac {1}{2} \int x^2 \log ^2(2+x) \, dx-\frac {9}{16} \int x^2 \log ^2(2+x) \, dx-\frac {3}{4} \int x^2 \log (2+x) \log (3+x) \, dx+\frac {3}{4} \text {Subst}\left (\int x^2 \log ^3(x) \, dx,x,2+x\right )+2 \left (\frac {3}{2} \int x \log ^2(2+x) \, dx\right )+\frac {3}{2} \int x \log (2+x) \log (3+x) \, dx+\frac {3}{2} \text {Subst}\left (\int x^2 \log ^3(x) \, dx,x,2+x\right )+\frac {27}{16} \int x \log ^2(2+x) \, dx+2 \int x \log (2+x) \log (3+x) \, dx+2 \text {Subst}\left (\int \log ^3(x) \, dx,x,2+x\right )-\frac {9}{4} \text {Subst}\left (\int x \log ^3(x) \, dx,x,2+x\right )-3 \int \log (2+x) \log (3+x) \, dx+3 \int x \log (2+x) \log (3+x) \, dx+3 \text {Subst}\left (\int \log ^3(x) \, dx,x,2+x\right )-2 \left (3 \text {Subst}\left (\int x \log ^3(x) \, dx,x,2+x\right )\right )-4 \int \log (2+x) \log (3+x) \, dx-2 \left (\frac {9}{2} \int \log ^2(2+x) \, dx\right )+\frac {9}{2} \text {Subst}\left (\int \log ^3(x) \, dx,x,2+x\right )-\frac {81}{16} \int \log ^2(2+x) \, dx-6 \int \log ^2(2+x) \, dx-6 \int \log (2+x) \log (3+x) \, dx+6 \int \frac {\log (2+x) \log (3+x)}{2+x} \, dx+8 \int \frac {\log (2+x) \log (3+x)}{2+x} \, dx-12 \int \log (2+x) \log (3+x) \, dx+12 \int \frac {\log (2+x) \log (3+x)}{2+x} \, dx+2 \left (\frac {27}{2} \int \frac {\log ^2(2+x)}{3+x} \, dx\right )+\frac {243}{16} \int \frac {\log ^2(2+x)}{3+x} \, dx+18 \int \frac {\log ^2(2+x)}{3+x} \, dx+24 \int \frac {\log (2+x) \log (3+x)}{2+x} \, dx+24 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,2+x\right )+\frac {81}{2} \text {Subst}(\int \log (x) \, dx,x,2+x)+\frac {243}{2} \text {Subst}\left (\int \frac {\log (x) \operatorname {PolyLog}(2,-x)}{x} \, dx,x,2+x\right )-\int x^2 \log (2+x) \log (3+x) \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.68 \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\frac {-195984-558290 x+17705 x^2-1050 x^3+54 x^4+910528 \log (2+x)+400008 x \log (2+x)-22836 x^2 \log (2+x)+2072 x^3 \log (2+x)-162 x^4 \log (2+x)-302016 \log ^2(2+x)-118800 x \log ^2(2+x)+11880 x^2 \log ^2(2+x)-1680 x^3 \log ^2(2+x)+216 x^4 \log ^2(2+x)+48384 \log ^3(2+x)+15552 x \log ^3(2+x)-2592 x^2 \log ^3(2+x)+576 x^3 \log ^3(2+x)-144 x^4 \log ^3(2+x)+309078 \log (3+x)+79680 x \log (3+x)-5520 x^2 \log (3+x)+592 x^3 \log (3+x)-54 x^4 \log (3+x)-293976 \log (2+x) \log (3+x)-57600 x \log (2+x) \log (3+x)+7488 x^2 \log (2+x) \log (3+x)-1344 x^3 \log (2+x) \log (3+x)+216 x^4 \log (2+x) \log (3+x)+138672 \log ^2(2+x) \log (3+x)+13824 x \log ^2(2+x) \log (3+x)-3456 x^2 \log ^2(2+x) \log (3+x)+1152 x^3 \log ^2(2+x) \log (3+x)-432 x^4 \log ^2(2+x) \log (3+x)-46656 \log ^3(2+x) \log (3+x)+576 x^4 \log ^3(2+x) \log (3+x)-24 \left (5609-6756 \log (2+x)+4680 \log ^2(2+x)\right ) \operatorname {PolyLog}(2,-2-x)+288 (-563+780 \log (2+x)) \operatorname {PolyLog}(3,-2-x)-224640 \operatorname {PolyLog}(4,-2-x)}{2304} \]
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\[\int x^{3} \ln \left (2+x \right )^{3} \ln \left (3+x \right )d x\]
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\[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\int { x^{3} \log \left (x + 3\right ) \log \left (x + 2\right )^{3} \,d x } \]
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\[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\left (\frac {x^{4} \log {\left (x + 2 \right )}^{3}}{4} - \frac {3 x^{4} \log {\left (x + 2 \right )}^{2}}{16} + \frac {3 x^{4} \log {\left (x + 2 \right )}}{32} - \frac {3 x^{4}}{128} + \frac {x^{3} \log {\left (x + 2 \right )}^{2}}{2} - \frac {7 x^{3} \log {\left (x + 2 \right )}}{12} + \frac {37 x^{3}}{144} - \frac {3 x^{2} \log {\left (x + 2 \right )}^{2}}{2} + \frac {13 x^{2} \log {\left (x + 2 \right )}}{4} - \frac {115 x^{2}}{48} + 6 x \log {\left (x + 2 \right )}^{2} - 25 x \log {\left (x + 2 \right )} + \frac {415 x}{12} - 4 \log {\left (x + 2 \right )}^{3} + 25 \log {\left (x + 2 \right )}^{2} - \frac {415 \log {\left (x + 2 \right )}}{6} + \frac {10955281}{240000}\right ) \log {\left (x + 3 \right )} - \frac {\int \frac {24900000 x}{x + 3}\, dx + \int \left (- \frac {1725000 x^{2}}{x + 3}\right )\, dx + \int \frac {185000 x^{3}}{x + 3}\, dx + \int \left (- \frac {16875 x^{4}}{x + 3}\right )\, dx + \int \left (- \frac {49800000 \log {\left (x + 2 \right )}}{x + 3}\right )\, dx + \int \frac {18000000 \log {\left (x + 2 \right )}^{2}}{x + 3}\, dx + \int \left (- \frac {2880000 \log {\left (x + 2 \right )}^{3}}{x + 3}\right )\, dx + \int \left (- \frac {18000000 x \log {\left (x + 2 \right )}}{x + 3}\right )\, dx + \int \frac {4320000 x \log {\left (x + 2 \right )}^{2}}{x + 3}\, dx + \int \frac {2340000 x^{2} \log {\left (x + 2 \right )}}{x + 3}\, dx + \int \left (- \frac {1080000 x^{2} \log {\left (x + 2 \right )}^{2}}{x + 3}\right )\, dx + \int \left (- \frac {420000 x^{3} \log {\left (x + 2 \right )}}{x + 3}\right )\, dx + \int \frac {360000 x^{3} \log {\left (x + 2 \right )}^{2}}{x + 3}\, dx + \int \frac {67500 x^{4} \log {\left (x + 2 \right )}}{x + 3}\, dx + \int \left (- \frac {135000 x^{4} \log {\left (x + 2 \right )}^{2}}{x + 3}\right )\, dx + \int \frac {180000 x^{4} \log {\left (x + 2 \right )}^{3}}{x + 3}\, dx + \int \frac {32865843}{x + 3}\, dx}{720000} \]
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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) \]
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\[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\int { x^{3} \log \left (x + 3\right ) \log \left (x + 2\right )^{3} \,d x } \]
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Timed out. \[ \int x^3 \log ^3(2+x) \log (3+x) \, dx=\int x^3\,{\ln \left (x+2\right )}^3\,\ln \left (x+3\right ) \,d x \]
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