Integrate[x^2*Log[a + b*x]*Log[c + d*x]^2, x] ==
(-132*a*b^2*c^2*d - 72*a^2*b*c*d^2 -
274*b^3*c^2*d*x - 132*a*b^2*c*d^2*x -
104*a^2*b*d^3*x + 38*b^3*c*d^2*x^2 +
19*a*b^2*d^3*x^2 - 8*b^3*d^3*x^3 +
132*a*b^2*c^2*d*Log[a + b*x] +
30*a^2*b*c*d^2*Log[a + b*x] +
8*a^3*d^3*Log[a + b*x] + 132*b^3*c^2*d*x*
Log[a + b*x] - 30*b^3*c*d^2*x^2*Log[a + b*x] +
8*b^3*d^3*x^3*Log[a + b*x] +
142*b^3*c^3*Log[c + d*x] + 102*a*b^2*c^2*d*
Log[c + d*x] + 96*a^2*b*c*d^2*Log[c + d*x] +
96*b^3*c^2*d*x*Log[c + d*x] + 72*a*b^2*c*d^2*x*
Log[c + d*x] + 96*a^2*b*d^3*x*Log[c + d*x] -
30*b^3*c*d^2*x^2*Log[c + d*x] -
30*a*b^2*d^3*x^2*Log[c + d*x] +
16*b^3*d^3*x^3*Log[c + d*x] -
72*a*b^2*c^2*d*Log[a + b*x]*Log[c + d*x] -
36*a^2*b*c*d^2*Log[a + b*x]*Log[c + d*x] -
24*a^3*d^3*Log[a + b*x]*Log[c + d*x] -
72*b^3*c^2*d*x*Log[a + b*x]*Log[c + d*x] +
36*b^3*c*d^2*x^2*Log[a + b*x]*Log[c + d*x] -
24*b^3*d^3*x^3*Log[a + b*x]*Log[c + d*x] -
12*b^3*c^3*Log[c + d*x]^2 - 18*a*b^2*c^2*d*
Log[c + d*x]^2 - 36*a^2*b*c*d^2*Log[c + d*x]^2 -
36*a^2*b*d^3*x*Log[c + d*x]^2 +
18*a*b^2*d^3*x^2*Log[c + d*x]^2 -
12*b^3*d^3*x^3*Log[c + d*x]^2 +
36*b^3*c^3*Log[a + b*x]*Log[c + d*x]^2 +
36*b^3*d^3*x^3*Log[a + b*x]*Log[c + d*x]^2 -
36*b^3*c^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*
Log[c + d*x]^2 + 36*a^3*d^3*
Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 -
132*b^3*c^3*Log[a + b*x]*
Log[(b*(c + d*x))/(b*c - a*d)] +
72*a*b^2*c^2*d*Log[a + b*x]*
Log[(b*(c + d*x))/(b*c - a*d)] +
36*a^2*b*c*d^2*Log[a + b*x]*
Log[(b*(c + d*x))/(b*c - a*d)] +
24*a^3*d^3*Log[a + b*x]*
Log[(b*(c + d*x))/(b*c - a*d)] +
12*(-11*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 +
2*a^3*d^3)*PolyLog[2, (d*(a + b*x))/
(-(b*c) + a*d)] - 72*(b^3*c^3 - a^3*d^3)*
Log[c + d*x]*PolyLog[2, (b*(c + d*x))/
(b*c - a*d)] + 72*b^3*c^3*
PolyLog[3, (b*(c + d*x))/(b*c - a*d)] -
72*a^3*d^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/
(108*b^3*d^3)
2 2 Integrate[x Log[a + b x] Log[c + d x] , x] ==
2 2 2 2 3 2
(-132 a b c d - 72 a b c d - 274 b c d x -
2 2 2 3 3 2 2
132 a b c d x - 104 a b d x + 38 b c d x +
2 3 2 3 3 3
19 a b d x - 8 b d x +
2 2
132 a b c d Log[a + b x] +
2 2
30 a b c d Log[a + b x] +
3 3
8 a d Log[a + b x] +
3 2
132 b c d x Log[a + b x] -
3 2 2
30 b c d x Log[a + b x] +
3 3 3
8 b d x Log[a + b x] +
3 3
142 b c Log[c + d x] +
2 2
102 a b c d Log[c + d x] +
2 2
96 a b c d Log[c + d x] +
3 2
96 b c d x Log[c + d x] +
2 2
72 a b c d x Log[c + d x] +
2 3
96 a b d x Log[c + d x] -
3 2 2
30 b c d x Log[c + d x] -
2 3 2
30 a b d x Log[c + d x] +
3 3 3
16 b d x Log[c + d x] -
2 2
72 a b c d Log[a + b x] Log[c + d x] -
2 2
36 a b c d Log[a + b x] Log[c + d x] -
3 3
24 a d Log[a + b x] Log[c + d x] -
3 2
72 b c d x Log[a + b x] Log[c + d x] +
3 2 2
36 b c d x Log[a + b x] Log[c + d x] -
3 3 3
24 b d x Log[a + b x] Log[c + d x] -
3 3 2
12 b c Log[c + d x] -
2 2 2
18 a b c d Log[c + d x] -
2 2 2
36 a b c d Log[c + d x] -
2 3 2
36 a b d x Log[c + d x] +
2 3 2 2
18 a b d x Log[c + d x] -
3 3 3 2
12 b d x Log[c + d x] +
3 3 2
36 b c Log[a + b x] Log[c + d x] +
3 3 3 2
36 b d x Log[a + b x] Log[c + d x] -
3 3 d (a + b x) 2
36 b c Log[------------] Log[c + d x] +
-(b c) + a d
3 3 d (a + b x) 2
36 a d Log[------------] Log[c + d x] -
-(b c) + a d
3 3 b (c + d x)
132 b c Log[a + b x] Log[-----------] +
b c - a d
2 2 b (c + d x)
72 a b c d Log[a + b x] Log[-----------] +
b c - a d
2 2 b (c + d x)
36 a b c d Log[a + b x] Log[-----------] +
b c - a d
3 3 b (c + d x)
24 a d Log[a + b x] Log[-----------] +
b c - a d
3 3 2 2 2 2
12 (-11 b c + 6 a b c d + 3 a b c d +
3 3 d (a + b x)
2 a d ) PolyLog[2, ------------] -
-(b c) + a d
3 3 3 3
72 (b c - a d ) Log[c + d x]
b (c + d x)
PolyLog[2, -----------] +
b c - a d
3 3 b (c + d x)
72 b c PolyLog[3, -----------] -
b c - a d
3 3 b (c + d x) 3 3
72 a d PolyLog[3, -----------]) / (108 b d )
b c - a d