Integrand size = 17, antiderivative size = 385 \[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=-\frac {(b d-a e)^2 x}{3 b^2}-\frac {(b d-a e) (b c d+e-a c e) x}{6 b^2 c}-\frac {(b c d+e-a c e)^2 x}{9 b^2 c^2}-\frac {(b d-a e) (d+e x)^2}{12 b e}-\frac {(b c d+e-a c e) (d+e x)^2}{18 b c e}-\frac {(d+e x)^3}{27 e}-\frac {(b d-a e) (b c d+e-a c e)^2 \log (1-a c-b c x)}{6 b^3 c^2 e}-\frac {(b c d+e-a c e)^3 \log (1-a c-b c x)}{9 b^3 c^3 e}-\frac {(b d-a e)^2 (1-a c-b c x) \log (1-a c-b c x)}{3 b^3 c}+\frac {(b d-a e) (d+e x)^2 \log (1-a c-b c x)}{6 b e}+\frac {(d+e x)^3 \log (1-a c-b c x)}{9 e}-\frac {(b d-a e)^3 \operatorname {PolyLog}(2,c (a+b x))}{3 b^3 e}+\frac {(d+e x)^3 \operatorname {PolyLog}(2,c (a+b x))}{3 e} \]
-1/3*(-a*e+b*d)^2*x/b^2-1/6*(-a*e+b*d)*(-a*c*e+b*c*d+e)*x/b^2/c-1/9*(-a*c* e+b*c*d+e)^2*x/b^2/c^2-1/12*(-a*e+b*d)*(e*x+d)^2/b/e-1/18*(-a*c*e+b*c*d+e) *(e*x+d)^2/b/c/e-1/27*(e*x+d)^3/e-1/6*(-a*e+b*d)*(-a*c*e+b*c*d+e)^2*ln(-b* c*x-a*c+1)/b^3/c^2/e-1/9*(-a*c*e+b*c*d+e)^3*ln(-b*c*x-a*c+1)/b^3/c^3/e-1/3 *(-a*e+b*d)^2*(-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)/b^3/c+1/6*(-a*e+b*d)*(e*x+d) ^2*ln(-b*c*x-a*c+1)/b/e+1/9*(e*x+d)^3*ln(-b*c*x-a*c+1)/e-1/3*(-a*e+b*d)^3* polylog(2,c*(b*x+a))/b^3/e+1/3*(e*x+d)^3*polylog(2,c*(b*x+a))/e
Time = 0.14 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.71 \[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=\frac {6 e (-1+a c+b c x) \left (\left (2-7 a c+11 a^2 c^2\right ) e+b^2 c^2 x (9 d+2 e x)+b c ((9-27 a c) d+(2-5 a c) e x)\right ) \log (1-a c-b c x)+b c \left (-66 a^2 c^2 e^2 x-x \left (12 e^2+6 b c e (9 d+e x)+b^2 c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )+3 a c \left (14 e^2 x+b c \left (-36 d^2+54 d e x+5 e^2 x^2\right )\right )+108 b c d^2 (-1+a c+b c x) \log (1-c (a+b x))\right )+36 c^3 \left (3 a b^2 d^2-3 a^2 b d e+a^3 e^2+b^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{108 b^3 c^3} \]
(6*e*(-1 + a*c + b*c*x)*((2 - 7*a*c + 11*a^2*c^2)*e + b^2*c^2*x*(9*d + 2*e *x) + b*c*((9 - 27*a*c)*d + (2 - 5*a*c)*e*x))*Log[1 - a*c - b*c*x] + b*c*( -66*a^2*c^2*e^2*x - x*(12*e^2 + 6*b*c*e*(9*d + e*x) + b^2*c^2*(108*d^2 + 2 7*d*e*x + 4*e^2*x^2)) + 3*a*c*(14*e^2*x + b*c*(-36*d^2 + 54*d*e*x + 5*e^2* x^2)) + 108*b*c*d^2*(-1 + a*c + b*c*x)*Log[1 - c*(a + b*x)]) + 36*c^3*(3*a *b^2*d^2 - 3*a^2*b*d*e + a^3*e^2 + b^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*Poly Log[2, c*(a + b*x)])/(108*b^3*c^3)
Time = 0.64 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {7152, 2865, 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.
| \(\displaystyle \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx\) |
| \(\Big \downarrow \) 7152 |
| \(\displaystyle \frac {b \int \frac {(d+e x)^3 \log (-a c-b x c+1)}{a+b x}dx}{3 e}+\frac {(d+e x)^3 \operatorname {PolyLog}(2,c (a+b x))}{3 e}\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle \frac {b \int \left (\frac {\log (-a c-b x c+1) (b d-a e)^3}{b^3 (a+b x)}+\frac {e \log (-a c-b x c+1) (b d-a e)^2}{b^3}+\frac {e (d+e x) \log (-a c-b x c+1) (b d-a e)}{b^2}+\frac {e (d+e x)^2 \log (-a c-b x c+1)}{b}\right )dx}{3 e}+\frac {(d+e x)^3 \operatorname {PolyLog}(2,c (a+b x))}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (-\frac {(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{3 b^4 c^3}-\frac {(b d-a e) (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{2 b^4 c^2}-\frac {(b d-a e)^3 \operatorname {PolyLog}(2,c (a+b x))}{b^4}-\frac {e (-a c-b c x+1) (b d-a e)^2 \log (-a c-b c x+1)}{b^4 c}-\frac {e x (-a c e+b c d+e)^2}{3 b^3 c^2}-\frac {e x (b d-a e) (-a c e+b c d+e)}{2 b^3 c}-\frac {e x (b d-a e)^2}{b^3}-\frac {(d+e x)^2 (-a c e+b c d+e)}{6 b^2 c}+\frac {(d+e x)^2 (b d-a e) \log (-a c-b c x+1)}{2 b^2}-\frac {(d+e x)^2 (b d-a e)}{4 b^2}+\frac {(d+e x)^3 \log (-a c-b c x+1)}{3 b}-\frac {(d+e x)^3}{9 b}\right )}{3 e}+\frac {(d+e x)^3 \operatorname {PolyLog}(2,c (a+b x))}{3 e}\) |
((d + e*x)^3*PolyLog[2, c*(a + b*x)])/(3*e) + (b*(-((e*(b*d - a*e)^2*x)/b^ 3) - (e*(b*d - a*e)*(b*c*d + e - a*c*e)*x)/(2*b^3*c) - (e*(b*c*d + e - a*c *e)^2*x)/(3*b^3*c^2) - ((b*d - a*e)*(d + e*x)^2)/(4*b^2) - ((b*c*d + e - a *c*e)*(d + e*x)^2)/(6*b^2*c) - (d + e*x)^3/(9*b) - ((b*d - a*e)*(b*c*d + e - a*c*e)^2*Log[1 - a*c - b*c*x])/(2*b^4*c^2) - ((b*c*d + e - a*c*e)^3*Log [1 - a*c - b*c*x])/(3*b^4*c^3) - (e*(b*d - a*e)^2*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b^4*c) + ((b*d - a*e)*(d + e*x)^2*Log[1 - a*c - b*c*x])/( 2*b^2) + ((d + e*x)^3*Log[1 - a*c - b*c*x])/(3*b) - ((b*d - a*e)^3*PolyLog [2, c*(a + b*x)])/b^4))/(3*e)
3.2.38.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
Int[((d_.) + (e_.)*(x_))^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Sy mbol] :> Simp[(d + e*x)^(m + 1)*(PolyLog[2, c*(a + b*x)]/(e*(m + 1))), x] + Simp[b/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(Log[1 - a*c - b*c*x]/(a + b*x) ), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 1.83 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.50
| method | result | size |
| parts | \(\frac {\operatorname {polylog}\left (2, c \left (b x +a \right )\right ) e^{2} x^{3}}{3}+\operatorname {polylog}\left (2, c \left (b x +a \right )\right ) e d \,x^{2}+\operatorname {polylog}\left (2, c \left (b x +a \right )\right ) d^{2} x +\frac {\operatorname {polylog}\left (2, c \left (b x +a \right )\right ) d^{3}}{3 e}-\frac {\frac {e \left (3 a^{2} c^{2} e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-6 a b \,c^{2} d e \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 b^{2} c^{2} d^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 a c \,e^{2} \left (\frac {\left (-b c x -a c +1\right )^{2} \ln \left (-b c x -a c +1\right )}{2}-\frac {\left (-b c x -a c +1\right )^{2}}{4}\right )-3 b c d e \left (\frac {\left (-b c x -a c +1\right )^{2} \ln \left (-b c x -a c +1\right )}{2}-\frac {\left (-b c x -a c +1\right )^{2}}{4}\right )-3 a c \,e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 b c d e \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+e^{2} \left (\frac {\left (-b c x -a c +1\right )^{3} \ln \left (-b c x -a c +1\right )}{3}-\frac {\left (-b c x -a c +1\right )^{3}}{9}\right )-2 e^{2} \left (\frac {\left (-b c x -a c +1\right )^{2} \ln \left (-b c x -a c +1\right )}{2}-\frac {\left (-b c x -a c +1\right )^{2}}{4}\right )+e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )\right )}{b^{3} c^{2}}-\frac {c \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \operatorname {dilog}\left (-b c x -a c +1\right )}{b^{3}}}{3 e c}\) | \(577\) |
| parallelrisch | \(\frac {-12 e^{2}-108 \operatorname {polylog}\left (2, c \left (b x +a \right )\right ) a^{2} b \,c^{3} d e -162 \ln \left (1-c \left (b x +a \right )\right ) a^{2} b \,c^{3} d e +216 \ln \left (1-c \left (b x +a \right )\right ) a b \,c^{2} d e +36 x^{3} \operatorname {polylog}\left (2, c \left (b x +a \right )\right ) b^{3} c^{3} e^{2}+12 x^{3} \ln \left (1-c \left (b x +a \right )\right ) b^{3} c^{3} e^{2}+15 x^{2} a \,b^{2} c^{3} e^{2}-27 x^{2} b^{3} c^{3} d e +108 x \operatorname {polylog}\left (2, c \left (b x +a \right )\right ) b^{3} c^{3} d^{2}+108 x \ln \left (1-c \left (b x +a \right )\right ) b^{3} c^{3} d^{2}-66 x \,a^{2} b \,c^{3} e^{2}+108 \operatorname {polylog}\left (2, c \left (b x +a \right )\right ) a \,b^{2} c^{3} d^{2}+108 \ln \left (1-c \left (b x +a \right )\right ) a \,b^{2} c^{3} d^{2}+42 e^{2} a b x \,c^{2}+216 a \,b^{2} c^{3} d^{2}-297 a^{2} b \,c^{3} d e +243 a b \,c^{2} d e -54 x \,b^{2} c^{2} d e -54 \ln \left (1-c \left (b x +a \right )\right ) b c d e +108 x^{2} \operatorname {polylog}\left (2, c \left (b x +a \right )\right ) b^{3} c^{3} d e -18 x^{2} \ln \left (1-c \left (b x +a \right )\right ) a \,b^{2} c^{3} e^{2}+54 x^{2} \ln \left (1-c \left (b x +a \right )\right ) b^{3} c^{3} d e +36 x \ln \left (1-c \left (b x +a \right )\right ) a^{2} b \,c^{3} e^{2}+162 x a \,b^{2} c^{3} d e -108 b^{2} c^{2} d^{2}-129 a^{2} c^{2} e^{2}-108 x \ln \left (1-c \left (b x +a \right )\right ) a \,b^{2} c^{3} d e -4 x^{3} b^{3} c^{3} e^{2}-108 x \,b^{3} c^{3} d^{2}+36 \operatorname {polylog}\left (2, c \left (b x +a \right )\right ) a^{3} c^{3} e^{2}+66 \ln \left (1-c \left (b x +a \right )\right ) a^{3} c^{3} e^{2}-6 x^{2} b^{2} c^{2} e^{2}-108 \ln \left (1-c \left (b x +a \right )\right ) a^{2} c^{2} e^{2}-108 \ln \left (1-c \left (b x +a \right )\right ) b^{2} c^{2} d^{2}-12 x b c \,e^{2}+54 \ln \left (1-c \left (b x +a \right )\right ) a c \,e^{2}-12 \ln \left (1-c \left (b x +a \right )\right ) e^{2}+117 a^{3} c^{3} e^{2}+60 a c \,e^{2}-54 b c d e}{108 b^{3} c^{3}}\) | \(670\) |
| derivativedivides | \(\frac {-\frac {c \,e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) a^{3}}{3 b^{2}}+\frac {c e \operatorname {polylog}\left (2, b c x +a c \right ) a^{2} d}{b}-c \operatorname {polylog}\left (2, b c x +a c \right ) a \,d^{2}+\frac {b c \operatorname {polylog}\left (2, b c x +a c \right ) d^{3}}{3 e}+\frac {e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) a^{2} \left (b c x +a c \right )}{b^{2}}-\frac {2 e \operatorname {polylog}\left (2, b c x +a c \right ) a d \left (b c x +a c \right )}{b}+\operatorname {polylog}\left (2, b c x +a c \right ) d^{2} \left (b c x +a c \right )-\frac {e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) a \left (b c x +a c \right )^{2}}{b^{2} c}+\frac {e \operatorname {polylog}\left (2, b c x +a c \right ) d \left (b c x +a c \right )^{2}}{b c}+\frac {e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) \left (b c x +a c \right )^{3}}{3 b^{2} c^{2}}-\frac {e^{3} \left (\frac {\left (-b c x -a c +1\right )^{3} \ln \left (-b c x -a c +1\right )}{3}-\frac {\left (-b c x -a c +1\right )^{3}}{9}\right )-\left (-3 a \,e^{3} c +3 b c d \,e^{2}+2 e^{3}\right ) \left (\frac {\left (-b c x -a c +1\right )^{2} \ln \left (-b c x -a c +1\right )}{2}-\frac {\left (-b c x -a c +1\right )^{2}}{4}\right )+3 a^{2} c^{2} e^{3} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-6 a b \,c^{2} d \,e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 b^{2} c^{2} d^{2} e \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-3 a c \,e^{3} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 b c d \,e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+e^{3} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-c^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \operatorname {dilog}\left (-b c x -a c +1\right )}{3 b^{2} c^{2} e}}{b c}\) | \(685\) |
| default | \(\frac {-\frac {c \,e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) a^{3}}{3 b^{2}}+\frac {c e \operatorname {polylog}\left (2, b c x +a c \right ) a^{2} d}{b}-c \operatorname {polylog}\left (2, b c x +a c \right ) a \,d^{2}+\frac {b c \operatorname {polylog}\left (2, b c x +a c \right ) d^{3}}{3 e}+\frac {e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) a^{2} \left (b c x +a c \right )}{b^{2}}-\frac {2 e \operatorname {polylog}\left (2, b c x +a c \right ) a d \left (b c x +a c \right )}{b}+\operatorname {polylog}\left (2, b c x +a c \right ) d^{2} \left (b c x +a c \right )-\frac {e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) a \left (b c x +a c \right )^{2}}{b^{2} c}+\frac {e \operatorname {polylog}\left (2, b c x +a c \right ) d \left (b c x +a c \right )^{2}}{b c}+\frac {e^{2} \operatorname {polylog}\left (2, b c x +a c \right ) \left (b c x +a c \right )^{3}}{3 b^{2} c^{2}}-\frac {e^{3} \left (\frac {\left (-b c x -a c +1\right )^{3} \ln \left (-b c x -a c +1\right )}{3}-\frac {\left (-b c x -a c +1\right )^{3}}{9}\right )-\left (-3 a \,e^{3} c +3 b c d \,e^{2}+2 e^{3}\right ) \left (\frac {\left (-b c x -a c +1\right )^{2} \ln \left (-b c x -a c +1\right )}{2}-\frac {\left (-b c x -a c +1\right )^{2}}{4}\right )+3 a^{2} c^{2} e^{3} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-6 a b \,c^{2} d \,e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 b^{2} c^{2} d^{2} e \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-3 a c \,e^{3} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+3 b c d \,e^{2} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )+e^{3} \left (\left (-b c x -a c +1\right ) \ln \left (-b c x -a c +1\right )-1+b c x +a c \right )-c^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \operatorname {dilog}\left (-b c x -a c +1\right )}{3 b^{2} c^{2} e}}{b c}\) | \(685\) |
1/3*polylog(2,c*(b*x+a))*e^2*x^3+polylog(2,c*(b*x+a))*e*d*x^2+polylog(2,c* (b*x+a))*d^2*x+1/3*polylog(2,c*(b*x+a))/e*d^3-1/3/e/c*(e/b^3/c^2*(3*a^2*c^ 2*e^2*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+b*c*x+a*c)-6*a*b*c^2*d*e*((-b*c*x -a*c+1)*ln(-b*c*x-a*c+1)-1+b*c*x+a*c)+3*b^2*c^2*d^2*((-b*c*x-a*c+1)*ln(-b* c*x-a*c+1)-1+b*c*x+a*c)+3*a*c*e^2*(1/2*(-b*c*x-a*c+1)^2*ln(-b*c*x-a*c+1)-1 /4*(-b*c*x-a*c+1)^2)-3*b*c*d*e*(1/2*(-b*c*x-a*c+1)^2*ln(-b*c*x-a*c+1)-1/4* (-b*c*x-a*c+1)^2)-3*a*c*e^2*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+b*c*x+a*c)+ 3*b*c*d*e*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+b*c*x+a*c)+e^2*(1/3*(-b*c*x-a *c+1)^3*ln(-b*c*x-a*c+1)-1/9*(-b*c*x-a*c+1)^3)-2*e^2*(1/2*(-b*c*x-a*c+1)^2 *ln(-b*c*x-a*c+1)-1/4*(-b*c*x-a*c+1)^2)+e^2*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+ 1)-1+b*c*x+a*c))-c*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/b^3*dilog (-b*c*x-a*c+1))
Time = 0.24 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.97 \[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=-\frac {4 \, b^{3} c^{3} e^{2} x^{3} + 3 \, {\left (9 \, b^{3} c^{3} d e - {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (18 \, b^{3} c^{3} d^{2} - 9 \, {\left (3 \, a b^{2} c^{3} - b^{2} c^{2}\right )} d e + {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} e^{2}\right )} x - 36 \, {\left (b^{3} c^{3} e^{2} x^{3} + 3 \, b^{3} c^{3} d e x^{2} + 3 \, b^{3} c^{3} d^{2} x + 3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{3} d e + a^{3} c^{3} e^{2}\right )} {\rm Li}_2\left (b c x + a c\right ) - 6 \, {\left (2 \, b^{3} c^{3} e^{2} x^{3} + 18 \, {\left (a b^{2} c^{3} - b^{2} c^{2}\right )} d^{2} - 9 \, {\left (3 \, a^{2} b c^{3} - 4 \, a b c^{2} + b c\right )} d e + {\left (11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} e^{2} + 3 \, {\left (3 \, b^{3} c^{3} d e - a b^{2} c^{3} e^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{3} d e + a^{2} b c^{3} e^{2}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \]
-1/108*(4*b^3*c^3*e^2*x^3 + 3*(9*b^3*c^3*d*e - (5*a*b^2*c^3 - 2*b^2*c^2)*e ^2)*x^2 + 6*(18*b^3*c^3*d^2 - 9*(3*a*b^2*c^3 - b^2*c^2)*d*e + (11*a^2*b*c^ 3 - 7*a*b*c^2 + 2*b*c)*e^2)*x - 36*(b^3*c^3*e^2*x^3 + 3*b^3*c^3*d*e*x^2 + 3*b^3*c^3*d^2*x + 3*a*b^2*c^3*d^2 - 3*a^2*b*c^3*d*e + a^3*c^3*e^2)*dilog(b *c*x + a*c) - 6*(2*b^3*c^3*e^2*x^3 + 18*(a*b^2*c^3 - b^2*c^2)*d^2 - 9*(3*a ^2*b*c^3 - 4*a*b*c^2 + b*c)*d*e + (11*a^3*c^3 - 18*a^2*c^2 + 9*a*c - 2)*e^ 2 + 3*(3*b^3*c^3*d*e - a*b^2*c^3*e^2)*x^2 + 6*(3*b^3*c^3*d^2 - 3*a*b^2*c^3 *d*e + a^2*b*c^3*e^2)*x)*log(-b*c*x - a*c + 1))/(b^3*c^3)
Time = 7.00 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.46 \[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=\begin {cases} 0 & \text {for}\: b = 0 \wedge c = 0 \\\left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) \operatorname {Li}_{2}\left (a c\right ) & \text {for}\: b = 0 \\0 & \text {for}\: c = 0 \\- \frac {11 a^{3} e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{18 b^{3}} + \frac {a^{3} e^{2} \operatorname {Li}_{2}\left (a c + b c x\right )}{3 b^{3}} + \frac {3 a^{2} d e \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{2}} - \frac {a^{2} d e \operatorname {Li}_{2}\left (a c + b c x\right )}{b^{2}} - \frac {a^{2} e^{2} x \operatorname {Li}_{1}\left (a c + b c x\right )}{3 b^{2}} - \frac {11 a^{2} e^{2} x}{18 b^{2}} + \frac {a^{2} e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{3} c} - \frac {a d^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b} + \frac {a d^{2} \operatorname {Li}_{2}\left (a c + b c x\right )}{b} + \frac {a d e x \operatorname {Li}_{1}\left (a c + b c x\right )}{b} + \frac {3 a d e x}{2 b} + \frac {a e^{2} x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{6 b} + \frac {5 a e^{2} x^{2}}{36 b} - \frac {2 a d e \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{2} c} + \frac {7 a e^{2} x}{18 b^{2} c} - \frac {a e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{3} c^{2}} - d^{2} x \operatorname {Li}_{1}\left (a c + b c x\right ) + d^{2} x \operatorname {Li}_{2}\left (a c + b c x\right ) - d^{2} x - \frac {d e x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{2} + d e x^{2} \operatorname {Li}_{2}\left (a c + b c x\right ) - \frac {d e x^{2}}{4} - \frac {e^{2} x^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{9} + \frac {e^{2} x^{3} \operatorname {Li}_{2}\left (a c + b c x\right )}{3} - \frac {e^{2} x^{3}}{27} + \frac {d^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b c} - \frac {d e x}{2 b c} - \frac {e^{2} x^{2}}{18 b c} + \frac {d e \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{2} c^{2}} - \frac {e^{2} x}{9 b^{2} c^{2}} + \frac {e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{9 b^{3} c^{3}} & \text {otherwise} \end {cases} \]
Piecewise((0, Eq(b, 0) & Eq(c, 0)), ((d**2*x + d*e*x**2 + e**2*x**3/3)*pol ylog(2, a*c), Eq(b, 0)), (0, Eq(c, 0)), (-11*a**3*e**2*polylog(1, a*c + b* c*x)/(18*b**3) + a**3*e**2*polylog(2, a*c + b*c*x)/(3*b**3) + 3*a**2*d*e*p olylog(1, a*c + b*c*x)/(2*b**2) - a**2*d*e*polylog(2, a*c + b*c*x)/b**2 - a**2*e**2*x*polylog(1, a*c + b*c*x)/(3*b**2) - 11*a**2*e**2*x/(18*b**2) + a**2*e**2*polylog(1, a*c + b*c*x)/(b**3*c) - a*d**2*polylog(1, a*c + b*c*x )/b + a*d**2*polylog(2, a*c + b*c*x)/b + a*d*e*x*polylog(1, a*c + b*c*x)/b + 3*a*d*e*x/(2*b) + a*e**2*x**2*polylog(1, a*c + b*c*x)/(6*b) + 5*a*e**2* x**2/(36*b) - 2*a*d*e*polylog(1, a*c + b*c*x)/(b**2*c) + 7*a*e**2*x/(18*b* *2*c) - a*e**2*polylog(1, a*c + b*c*x)/(2*b**3*c**2) - d**2*x*polylog(1, a *c + b*c*x) + d**2*x*polylog(2, a*c + b*c*x) - d**2*x - d*e*x**2*polylog(1 , a*c + b*c*x)/2 + d*e*x**2*polylog(2, a*c + b*c*x) - d*e*x**2/4 - e**2*x* *3*polylog(1, a*c + b*c*x)/9 + e**2*x**3*polylog(2, a*c + b*c*x)/3 - e**2* x**3/27 + d**2*polylog(1, a*c + b*c*x)/(b*c) - d*e*x/(2*b*c) - e**2*x**2/( 18*b*c) + d*e*polylog(1, a*c + b*c*x)/(2*b**2*c**2) - e**2*x/(9*b**2*c**2) + e**2*polylog(1, a*c + b*c*x)/(9*b**3*c**3), True))
Time = 0.21 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.05 \[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=-\frac {{\left (3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2}\right )} {\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )}}{3 \, b^{3}} - \frac {4 \, b^{3} c^{3} e^{2} x^{3} + 3 \, {\left (9 \, b^{3} c^{3} d e - {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (18 \, b^{3} c^{3} d^{2} - 9 \, {\left (3 \, a b^{2} c^{3} - b^{2} c^{2}\right )} d e + {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} e^{2}\right )} x - 36 \, {\left (b^{3} c^{3} e^{2} x^{3} + 3 \, b^{3} c^{3} d e x^{2} + 3 \, b^{3} c^{3} d^{2} x\right )} {\rm Li}_2\left (b c x + a c\right ) - 6 \, {\left (2 \, b^{3} c^{3} e^{2} x^{3} + 18 \, {\left (a b^{2} c^{3} - b^{2} c^{2}\right )} d^{2} - 9 \, {\left (3 \, a^{2} b c^{3} - 4 \, a b c^{2} + b c\right )} d e + {\left (11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} e^{2} + 3 \, {\left (3 \, b^{3} c^{3} d e - a b^{2} c^{3} e^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{3} d e + a^{2} b c^{3} e^{2}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \]
-1/3*(3*a*b^2*d^2 - 3*a^2*b*d*e + a^3*e^2)*(log(b*c*x + a*c)*log(-b*c*x - a*c + 1) + dilog(-b*c*x - a*c + 1))/b^3 - 1/108*(4*b^3*c^3*e^2*x^3 + 3*(9* b^3*c^3*d*e - (5*a*b^2*c^3 - 2*b^2*c^2)*e^2)*x^2 + 6*(18*b^3*c^3*d^2 - 9*( 3*a*b^2*c^3 - b^2*c^2)*d*e + (11*a^2*b*c^3 - 7*a*b*c^2 + 2*b*c)*e^2)*x - 3 6*(b^3*c^3*e^2*x^3 + 3*b^3*c^3*d*e*x^2 + 3*b^3*c^3*d^2*x)*dilog(b*c*x + a* c) - 6*(2*b^3*c^3*e^2*x^3 + 18*(a*b^2*c^3 - b^2*c^2)*d^2 - 9*(3*a^2*b*c^3 - 4*a*b*c^2 + b*c)*d*e + (11*a^3*c^3 - 18*a^2*c^2 + 9*a*c - 2)*e^2 + 3*(3* b^3*c^3*d*e - a*b^2*c^3*e^2)*x^2 + 6*(3*b^3*c^3*d^2 - 3*a*b^2*c^3*d*e + a^ 2*b*c^3*e^2)*x)*log(-b*c*x - a*c + 1))/(b^3*c^3)
\[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=\int { {\left (e x + d\right )}^{2} {\rm Li}_2\left ({\left (b x + a\right )} c\right ) \,d x } \]
Timed out. \[ \int (d+e x)^2 \operatorname {PolyLog}(2,c (a+b x)) \, dx=\int \mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]