3.29 \(\int x^3 \log ^3(2+x) \log (3+x) \, dx\)

Optimal. Leaf size=679 \[ -\frac{5609}{96} \text{PolyLog}(2,-x-2)-\frac{563}{8} \text{PolyLog}(3,-x-2)-\frac{195}{2} \text{PolyLog}(4,-x-2)-\frac{195}{4} \log ^2(x+2) \text{PolyLog}(2,-x-2)+\frac{563}{8} \log (x+2) \text{PolyLog}(2,-x-2)+\frac{195}{2} \log (x+2) \text{PolyLog}(3,-x-2)+\frac{3 x^4}{256}+\frac{1}{4} x^4 \log ^3(x+2) \log (x+3)+\frac{3}{64} x^4 \log ^2(x+2)-\frac{3}{16} x^4 \log ^2(x+2) \log (x+3)-\frac{3}{128} x^4 \log (x+2)+\frac{3}{32} x^4 \log (x+2) \log (x+3)-\frac{3}{128} x^4 \log (x+3)-\frac{763 x^3}{3456}-\frac{17}{48} x^3 \log ^2(x+2)+\frac{1}{2} x^3 \log ^2(x+2) \log (x+3)+\frac{83}{288} x^3 \log (x+2)-\frac{7}{12} x^3 \log (x+2) \log (x+3)+\frac{37}{144} x^3 \log (x+3)+\frac{8029 x^2}{2304}-\frac{3}{2} x^2 \log ^2(x+2) \log (x+3)-\frac{187}{64} x^2 \log (x+2)+\frac{13}{4} x^2 \log (x+2) \log (x+3)-\frac{115}{48} x^2 \log (x+3)-\frac{302177 x}{1152}+\frac{3}{256} (x+2)^4-\frac{71}{216} (x+2)^3+\frac{377}{64} (x+2)^2-\frac{1}{16} (x+2)^4 \log ^3(x+2)+\frac{3}{4} (x+2)^3 \log ^3(x+2)-\frac{33}{8} (x+2)^2 \log ^3(x+2)+\frac{65}{4} (x+2) \log ^3(x+2)-\frac{81}{4} \log ^3(x+2) \log (x+3)+6 x \log ^2(x+2) \log (x+3)+\frac{3}{64} (x+2)^4 \log ^2(x+2)-\frac{3}{4} (x+2)^3 \log ^2(x+2)+\frac{273}{32} (x+2)^2 \log ^2(x+2)-\frac{1251}{16} (x+2) \log ^2(x+2)+\frac{43}{12} \log ^2(x+2)+\frac{963}{16} \log ^2(x+2) \log (x+3)-25 x \log (x+2) \log (x+3)-\frac{3}{128} (x+2)^4 \log (x+2)+\frac{1}{2} (x+2)^3 \log (x+2)-\frac{273}{32} (x+2)^2 \log (x+2)+\frac{6365}{32} (x+2) \log (x+2)+\frac{1}{128} \left (-3 (x+2)^4+32 (x+2)^3-144 (x+2)^2+384 (x+2)-192 \log (x+2)\right ) \log (x+2)+\frac{17}{72} \left ((x+2)^3-9 (x+2)^2+36 (x+2)-24 \log (x+2)\right ) \log (x+2)+\frac{2069}{144} \log (x+2)+\frac{415}{12} (x+3) \log (x+3)-\frac{4083}{32} \log (x+2) \log (x+3)+\frac{3891}{128} \log (x+3) \]

[Out]

(-302177*x)/1152 + (8029*x^2)/2304 - (763*x^3)/3456 + (3*x^4)/256 + (377*(2 + x)
^2)/64 - (71*(2 + x)^3)/216 + (3*(2 + x)^4)/256 + (2069*Log[2 + x])/144 - (187*x
^2*Log[2 + x])/64 + (83*x^3*Log[2 + x])/288 - (3*x^4*Log[2 + x])/128 + (6365*(2
+ x)*Log[2 + x])/32 - (273*(2 + x)^2*Log[2 + x])/32 + ((2 + x)^3*Log[2 + x])/2 -
(3*(2 + x)^4*Log[2 + x])/128 + ((384*(2 + x) - 144*(2 + x)^2 + 32*(2 + x)^3 - 3
*(2 + x)^4 - 192*Log[2 + x])*Log[2 + x])/128 + (17*(36*(2 + x) - 9*(2 + x)^2 + (
2 + x)^3 - 24*Log[2 + x])*Log[2 + x])/72 + (43*Log[2 + x]^2)/12 - (17*x^3*Log[2
+ x]^2)/48 + (3*x^4*Log[2 + x]^2)/64 - (1251*(2 + x)*Log[2 + x]^2)/16 + (273*(2
+ x)^2*Log[2 + x]^2)/32 - (3*(2 + x)^3*Log[2 + x]^2)/4 + (3*(2 + x)^4*Log[2 + x]
^2)/64 + (65*(2 + x)*Log[2 + x]^3)/4 - (33*(2 + x)^2*Log[2 + x]^3)/8 + (3*(2 + x
)^3*Log[2 + x]^3)/4 - ((2 + x)^4*Log[2 + x]^3)/16 + (3891*Log[3 + x])/128 - (115
*x^2*Log[3 + x])/48 + (37*x^3*Log[3 + x])/144 - (3*x^4*Log[3 + x])/128 + (415*(3
+ x)*Log[3 + x])/12 - (4083*Log[2 + x]*Log[3 + x])/32 - 25*x*Log[2 + x]*Log[3 +
x] + (13*x^2*Log[2 + x]*Log[3 + x])/4 - (7*x^3*Log[2 + x]*Log[3 + x])/12 + (3*x
^4*Log[2 + x]*Log[3 + x])/32 + (963*Log[2 + x]^2*Log[3 + x])/16 + 6*x*Log[2 + x]
^2*Log[3 + x] - (3*x^2*Log[2 + x]^2*Log[3 + x])/2 + (x^3*Log[2 + x]^2*Log[3 + x]
)/2 - (3*x^4*Log[2 + x]^2*Log[3 + x])/16 - (81*Log[2 + x]^3*Log[3 + x])/4 + (x^4
*Log[2 + x]^3*Log[3 + x])/4 - (5609*PolyLog[2, -2 - x])/96 + (563*Log[2 + x]*Pol
yLog[2, -2 - x])/8 - (195*Log[2 + x]^2*PolyLog[2, -2 - x])/4 - (563*PolyLog[3, -
2 - x])/8 + (195*Log[2 + x]*PolyLog[3, -2 - x])/2 - (195*PolyLog[4, -2 - x])/2

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Rubi [A]  time = 7.49551, antiderivative size = 679, normalized size of antiderivative = 1., number of steps used = 359, number of rules used = 30, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.143 \[ -\frac{5609}{96} \text{PolyLog}(2,-x-2)-\frac{563}{8} \text{PolyLog}(3,-x-2)-\frac{195}{2} \text{PolyLog}(4,-x-2)-\frac{195}{4} \log ^2(x+2) \text{PolyLog}(2,-x-2)+\frac{563}{8} \log (x+2) \text{PolyLog}(2,-x-2)+\frac{195}{2} \log (x+2) \text{PolyLog}(3,-x-2)+\frac{3 x^4}{256}+\frac{1}{4} x^4 \log ^3(x+2) \log (x+3)+\frac{3}{64} x^4 \log ^2(x+2)-\frac{3}{16} x^4 \log ^2(x+2) \log (x+3)-\frac{3}{128} x^4 \log (x+2)+\frac{3}{32} x^4 \log (x+2) \log (x+3)-\frac{3}{128} x^4 \log (x+3)-\frac{763 x^3}{3456}-\frac{17}{48} x^3 \log ^2(x+2)+\frac{1}{2} x^3 \log ^2(x+2) \log (x+3)+\frac{83}{288} x^3 \log (x+2)-\frac{7}{12} x^3 \log (x+2) \log (x+3)+\frac{37}{144} x^3 \log (x+3)+\frac{8029 x^2}{2304}-\frac{3}{2} x^2 \log ^2(x+2) \log (x+3)-\frac{187}{64} x^2 \log (x+2)+\frac{13}{4} x^2 \log (x+2) \log (x+3)-\frac{115}{48} x^2 \log (x+3)-\frac{302177 x}{1152}+\frac{3}{256} (x+2)^4-\frac{71}{216} (x+2)^3+\frac{377}{64} (x+2)^2-\frac{1}{16} (x+2)^4 \log ^3(x+2)+\frac{3}{4} (x+2)^3 \log ^3(x+2)-\frac{33}{8} (x+2)^2 \log ^3(x+2)+\frac{65}{4} (x+2) \log ^3(x+2)-\frac{81}{4} \log ^3(x+2) \log (x+3)+6 x \log ^2(x+2) \log (x+3)+\frac{3}{64} (x+2)^4 \log ^2(x+2)-\frac{3}{4} (x+2)^3 \log ^2(x+2)+\frac{273}{32} (x+2)^2 \log ^2(x+2)-\frac{1251}{16} (x+2) \log ^2(x+2)+\frac{43}{12} \log ^2(x+2)+\frac{963}{16} \log ^2(x+2) \log (x+3)-25 x \log (x+2) \log (x+3)-\frac{3}{128} (x+2)^4 \log (x+2)+\frac{1}{2} (x+2)^3 \log (x+2)-\frac{273}{32} (x+2)^2 \log (x+2)+\frac{6365}{32} (x+2) \log (x+2)+\frac{1}{128} \left (-3 (x+2)^4+32 (x+2)^3-144 (x+2)^2+384 (x+2)-192 \log (x+2)\right ) \log (x+2)+\frac{17}{72} \left ((x+2)^3-9 (x+2)^2+36 (x+2)-24 \log (x+2)\right ) \log (x+2)+\frac{2069}{144} \log (x+2)+\frac{415}{12} (x+3) \log (x+3)-\frac{4083}{32} \log (x+2) \log (x+3)+\frac{3891}{128} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[x^3*Log[2 + x]^3*Log[3 + x],x]

[Out]

(-302177*x)/1152 + (8029*x^2)/2304 - (763*x^3)/3456 + (3*x^4)/256 + (377*(2 + x)
^2)/64 - (71*(2 + x)^3)/216 + (3*(2 + x)^4)/256 + (2069*Log[2 + x])/144 - (187*x
^2*Log[2 + x])/64 + (83*x^3*Log[2 + x])/288 - (3*x^4*Log[2 + x])/128 + (6365*(2
+ x)*Log[2 + x])/32 - (273*(2 + x)^2*Log[2 + x])/32 + ((2 + x)^3*Log[2 + x])/2 -
(3*(2 + x)^4*Log[2 + x])/128 + ((384*(2 + x) - 144*(2 + x)^2 + 32*(2 + x)^3 - 3
*(2 + x)^4 - 192*Log[2 + x])*Log[2 + x])/128 + (17*(36*(2 + x) - 9*(2 + x)^2 + (
2 + x)^3 - 24*Log[2 + x])*Log[2 + x])/72 + (43*Log[2 + x]^2)/12 - (17*x^3*Log[2
+ x]^2)/48 + (3*x^4*Log[2 + x]^2)/64 - (1251*(2 + x)*Log[2 + x]^2)/16 + (273*(2
+ x)^2*Log[2 + x]^2)/32 - (3*(2 + x)^3*Log[2 + x]^2)/4 + (3*(2 + x)^4*Log[2 + x]
^2)/64 + (65*(2 + x)*Log[2 + x]^3)/4 - (33*(2 + x)^2*Log[2 + x]^3)/8 + (3*(2 + x
)^3*Log[2 + x]^3)/4 - ((2 + x)^4*Log[2 + x]^3)/16 + (3891*Log[3 + x])/128 - (115
*x^2*Log[3 + x])/48 + (37*x^3*Log[3 + x])/144 - (3*x^4*Log[3 + x])/128 + (415*(3
+ x)*Log[3 + x])/12 - (4083*Log[2 + x]*Log[3 + x])/32 - 25*x*Log[2 + x]*Log[3 +
x] + (13*x^2*Log[2 + x]*Log[3 + x])/4 - (7*x^3*Log[2 + x]*Log[3 + x])/12 + (3*x
^4*Log[2 + x]*Log[3 + x])/32 + (963*Log[2 + x]^2*Log[3 + x])/16 + 6*x*Log[2 + x]
^2*Log[3 + x] - (3*x^2*Log[2 + x]^2*Log[3 + x])/2 + (x^3*Log[2 + x]^2*Log[3 + x]
)/2 - (3*x^4*Log[2 + x]^2*Log[3 + x])/16 - (81*Log[2 + x]^3*Log[3 + x])/4 + (x^4
*Log[2 + x]^3*Log[3 + x])/4 - (5609*PolyLog[2, -2 - x])/96 + (563*Log[2 + x]*Pol
yLog[2, -2 - x])/8 - (195*Log[2 + x]^2*PolyLog[2, -2 - x])/4 - (563*PolyLog[3, -
2 - x])/8 + (195*Log[2 + x]*PolyLog[3, -2 - x])/2 - (195*PolyLog[4, -2 - x])/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \log{\left (x + 2 \right )}^{3} \log{\left (x + 3 \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*ln(2+x)**3*ln(3+x),x)

[Out]

Integral(x**3*log(x + 2)**3*log(x + 3), x)

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Mathematica [A]  time = 0.266705, size = 412, normalized size = 0.61 \[ \frac{-224640 \text{PolyLog}(4,-x-2)-24 \left (4680 \log ^2(x+2)-6756 \log (x+2)+5609\right ) \text{PolyLog}(2,-x-2)+288 (780 \log (x+2)-563) \text{PolyLog}(3,-x-2)+54 x^4-144 x^4 \log ^3(x+2)+576 x^4 \log ^3(x+2) \log (x+3)+216 x^4 \log ^2(x+2)-432 x^4 \log ^2(x+2) \log (x+3)-162 x^4 \log (x+2)+216 x^4 \log (x+2) \log (x+3)-54 x^4 \log (x+3)-1050 x^3+576 x^3 \log ^3(x+2)-1680 x^3 \log ^2(x+2)+1152 x^3 \log ^2(x+2) \log (x+3)+2072 x^3 \log (x+2)-1344 x^3 \log (x+2) \log (x+3)+592 x^3 \log (x+3)+17705 x^2-2592 x^2 \log ^3(x+2)+11880 x^2 \log ^2(x+2)-3456 x^2 \log ^2(x+2) \log (x+3)-22836 x^2 \log (x+2)+7488 x^2 \log (x+2) \log (x+3)-5520 x^2 \log (x+3)-558290 x+15552 x \log ^3(x+2)+48384 \log ^3(x+2)-46656 \log ^3(x+2) \log (x+3)-118800 x \log ^2(x+2)+13824 x \log ^2(x+2) \log (x+3)-302016 \log ^2(x+2)+138672 \log ^2(x+2) \log (x+3)+400008 x \log (x+2)-57600 x \log (x+2) \log (x+3)+79680 x \log (x+3)+910528 \log (x+2)-293976 \log (x+2) \log (x+3)+309078 \log (x+3)-759744}{2304} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Log[2 + x]^3*Log[3 + x],x]

[Out]

(-759744 - 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]*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 + 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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( \ln \left ( 2+x \right ) \right ) ^{3}\ln \left ( 3+x \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*ln(2+x)^3*ln(3+x),x)

[Out]

int(x^3*ln(2+x)^3*ln(3+x),x)

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Maxima [A]  time = 1.39935, size = 699, normalized size = 1.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*log(x + 3)*log(x + 2)^3,x, algorithm="maxima")

[Out]

3/128*x^4 + 1/16*(4*x^4*log(x + 3) - x^4 + 4*x^3 - 18*x^2 + 108*x - 324*log(x +
3))*log(x + 2)^3 - 65/4*log(x + 3)*log(x + 2)^3 + 195/4*log(x + 3)*log(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 + 2
1*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 + 5
6160*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) + 17171/128*log(x + 3) + 14
227/36*log(x + 2) - 195/2*polylog(4, -x - 2) - 563/8*polylog(3, -x - 2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (x^{3} \log \left (x + 3\right ) \log \left (x + 2\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*log(x + 3)*log(x + 2)^3,x, algorithm="fricas")

[Out]

integral(x^3*log(x + 3)*log(x + 2)^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*ln(2+x)**3*ln(3+x),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \log \left (x + 3\right ) \log \left (x + 2\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*log(x + 3)*log(x + 2)^3,x, algorithm="giac")

[Out]

integrate(x^3*log(x + 3)*log(x + 2)^3, x)