3.195 \(\int \frac{(a+b x+c x^2) \log (1-d x) \text{PolyLog}(2,d x)}{x^2} \, dx\)

Optimal. Leaf size=218 \[ -2 d \left (a-\frac{c}{d^2}\right ) \text{PolyLog}(3,1-d x)+d \left (a-\frac{c}{d^2}\right ) \log (1-d x) \text{PolyLog}(2,d x)+2 d \left (a-\frac{c}{d^2}\right ) \log (1-d x) \text{PolyLog}(2,1-d x)-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{PolyLog}(2,d x)-2 a d \text{PolyLog}(2,d x)-a d \text{PolyLog}(3,d x)-\frac{1}{2} b \text{PolyLog}(2,d x)^2-c x \text{PolyLog}(2,d x)+d \left (a-\frac{c}{d^2}\right ) \log (d x) \log ^2(1-d x)+\frac{a (1-d x) \log ^2(1-d x)}{x}-\frac{c (1-d x) \log ^2(1-d x)}{d}+\frac{3 c (1-d x) \log (1-d x)}{d}+3 c x \]

[Out]

3*c*x + (3*c*(1 - d*x)*Log[1 - d*x])/d - (c*(1 - d*x)*Log[1 - d*x]^2)/d + (a*(1 - d*x)*Log[1 - d*x]^2)/x + (a
- c/d^2)*d*Log[d*x]*Log[1 - d*x]^2 - 2*a*d*PolyLog[2, d*x] - c*x*PolyLog[2, d*x] + (a - c/d^2)*d*Log[1 - d*x]*
PolyLog[2, d*x] - (a/x - c*x)*Log[1 - d*x]*PolyLog[2, d*x] - (b*PolyLog[2, d*x]^2)/2 + 2*(a - c/d^2)*d*Log[1 -
d*x]*PolyLog[2, 1 - d*x] - a*d*PolyLog[3, d*x] - 2*(a - c/d^2)*d*PolyLog[3, 1 - d*x]

________________________________________________________________________________________

Rubi [A]  time = 0.507956, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 21, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.808, Rules used = {6742, 6586, 2389, 2295, 6591, 2395, 36, 29, 31, 6589, 6605, 6601, 14, 6606, 2296, 2397, 2391, 6596, 2396, 2433, 2374} \[ -2 d \left (a-\frac{c}{d^2}\right ) \text{PolyLog}(3,1-d x)+d \left (a-\frac{c}{d^2}\right ) \log (1-d x) \text{PolyLog}(2,d x)+2 d \left (a-\frac{c}{d^2}\right ) \log (1-d x) \text{PolyLog}(2,1-d x)-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{PolyLog}(2,d x)-2 a d \text{PolyLog}(2,d x)-a d \text{PolyLog}(3,d x)-\frac{1}{2} b \text{PolyLog}(2,d x)^2-c x \text{PolyLog}(2,d x)+d \left (a-\frac{c}{d^2}\right ) \log (d x) \log ^2(1-d x)+\frac{a (1-d x) \log ^2(1-d x)}{x}-\frac{c (1-d x) \log ^2(1-d x)}{d}+\frac{3 c (1-d x) \log (1-d x)}{d}+3 c x \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x])/x^2,x]

[Out]

3*c*x + (3*c*(1 - d*x)*Log[1 - d*x])/d - (c*(1 - d*x)*Log[1 - d*x]^2)/d + (a*(1 - d*x)*Log[1 - d*x]^2)/x + (a
- c/d^2)*d*Log[d*x]*Log[1 - d*x]^2 - 2*a*d*PolyLog[2, d*x] - c*x*PolyLog[2, d*x] + (a - c/d^2)*d*Log[1 - d*x]*
PolyLog[2, d*x] - (a/x - c*x)*Log[1 - d*x]*PolyLog[2, d*x] - (b*PolyLog[2, d*x]^2)/2 + 2*(a - c/d^2)*d*Log[1 -
d*x]*PolyLog[2, 1 - d*x] - a*d*PolyLog[3, d*x] - 2*(a - c/d^2)*d*PolyLog[3, 1 - d*x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 6586

Int[PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[x*PolyLog[n, a*(b*x^p)^q], x] - Dist[p*q, I
nt[PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6605

Int[((g_.) + Log[1 + (e_.)*(x_)]*(h_.))*(Px_)*(x_)^(m_)*PolyLog[2, (c_.)*(x_)], x_Symbol] :> Dist[Coeff[Px, x,
-m - 1], Int[((g + h*Log[1 + e*x])*PolyLog[2, c*x])/x, x], x] + Int[x^m*(Px - Coeff[Px, x, -m - 1]*x^(-m - 1)
)*(g + h*Log[1 + e*x])*PolyLog[2, c*x], x] /; FreeQ[{c, e, g, h}, x] && PolyQ[Px, x] && ILtQ[m, 0] && EqQ[c +
e, 0] && NeQ[Coeff[Px, x, -m - 1], 0]

Rule 6601

Int[(Log[1 + (e_.)*(x_)]*PolyLog[2, (c_.)*(x_)])/(x_), x_Symbol] :> -Simp[PolyLog[2, c*x]^2/2, x] /; FreeQ[{c,
e}, x] && EqQ[c + e, 0]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6606

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(Px_)*(x_)^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_
))], x_Symbol] :> With[{u = IntHide[x^m*Px, x]}, Simp[u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x]
+ (Dist[b, Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x], x] - Dis
t[e*h*n, Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g,
h, n}, x] && PolyQ[Px, x] && IntegerQ[m]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6596

Int[PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 - a*c - b*c*x]*PolyL
og[2, c*(a + b*x)])/e, x] + Dist[b/e, Int[Log[1 - a*c - b*c*x]^2/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c*(b*d - a*e) + e, 0]

Rule 2396

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right ) \log (1-d x) \text{Li}_2(d x)}{x^2} \, dx &=b \int \frac{\log (1-d x) \text{Li}_2(d x)}{x} \, dx+\int \frac{\left (a+c x^2\right ) \log (1-d x) \text{Li}_2(d x)}{x^2} \, dx\\ &=-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2+d \int \left (-\frac{c \text{Li}_2(d x)}{d}-\frac{a \text{Li}_2(d x)}{x}+\frac{\left (-c+a d^2\right ) \text{Li}_2(d x)}{d (-1+d x)}\right ) \, dx+\int \left (c \log ^2(1-d x)-\frac{a \log ^2(1-d x)}{x^2}\right ) \, dx\\ &=-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2-a \int \frac{\log ^2(1-d x)}{x^2} \, dx+c \int \log ^2(1-d x) \, dx-c \int \text{Li}_2(d x) \, dx-(a d) \int \frac{\text{Li}_2(d x)}{x} \, dx+\left (-c+a d^2\right ) \int \frac{\text{Li}_2(d x)}{-1+d x} \, dx\\ &=\frac{a (1-d x) \log ^2(1-d x)}{x}-c x \text{Li}_2(d x)-\frac{\left (c-a d^2\right ) \log (1-d x) \text{Li}_2(d x)}{d}-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2-a d \text{Li}_3(d x)-c \int \log (1-d x) \, dx-\frac{c \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{d}+(2 a d) \int \frac{\log (1-d x)}{x} \, dx+\frac{\left (-c+a d^2\right ) \int \frac{\log ^2(1-d x)}{x} \, dx}{d}\\ &=-\frac{c (1-d x) \log ^2(1-d x)}{d}+\frac{a (1-d x) \log ^2(1-d x)}{x}-\frac{\left (c-a d^2\right ) \log (d x) \log ^2(1-d x)}{d}-2 a d \text{Li}_2(d x)-c x \text{Li}_2(d x)-\frac{\left (c-a d^2\right ) \log (1-d x) \text{Li}_2(d x)}{d}-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2-a d \text{Li}_3(d x)+\frac{c \operatorname{Subst}(\int \log (x) \, dx,x,1-d x)}{d}+\frac{(2 c) \operatorname{Subst}(\int \log (x) \, dx,x,1-d x)}{d}-\left (2 \left (c-a d^2\right )\right ) \int \frac{\log (d x) \log (1-d x)}{1-d x} \, dx\\ &=3 c x+\frac{3 c (1-d x) \log (1-d x)}{d}-\frac{c (1-d x) \log ^2(1-d x)}{d}+\frac{a (1-d x) \log ^2(1-d x)}{x}-\frac{\left (c-a d^2\right ) \log (d x) \log ^2(1-d x)}{d}-2 a d \text{Li}_2(d x)-c x \text{Li}_2(d x)-\frac{\left (c-a d^2\right ) \log (1-d x) \text{Li}_2(d x)}{d}-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2-a d \text{Li}_3(d x)+\frac{\left (2 \left (c-a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (d \left (\frac{1}{d}-\frac{x}{d}\right )\right )}{x} \, dx,x,1-d x\right )}{d}\\ &=3 c x+\frac{3 c (1-d x) \log (1-d x)}{d}-\frac{c (1-d x) \log ^2(1-d x)}{d}+\frac{a (1-d x) \log ^2(1-d x)}{x}-\frac{\left (c-a d^2\right ) \log (d x) \log ^2(1-d x)}{d}-2 a d \text{Li}_2(d x)-c x \text{Li}_2(d x)-\frac{\left (c-a d^2\right ) \log (1-d x) \text{Li}_2(d x)}{d}-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2-\frac{2 \left (c-a d^2\right ) \log (1-d x) \text{Li}_2(1-d x)}{d}-a d \text{Li}_3(d x)+\frac{\left (2 \left (c-a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-d x\right )}{d}\\ &=3 c x+\frac{3 c (1-d x) \log (1-d x)}{d}-\frac{c (1-d x) \log ^2(1-d x)}{d}+\frac{a (1-d x) \log ^2(1-d x)}{x}-\frac{\left (c-a d^2\right ) \log (d x) \log ^2(1-d x)}{d}-2 a d \text{Li}_2(d x)-c x \text{Li}_2(d x)-\frac{\left (c-a d^2\right ) \log (1-d x) \text{Li}_2(d x)}{d}-\left (\frac{a}{x}-c x\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{2} b \text{Li}_2(d x){}^2-\frac{2 \left (c-a d^2\right ) \log (1-d x) \text{Li}_2(1-d x)}{d}-a d \text{Li}_3(d x)+\frac{2 \left (c-a d^2\right ) \text{Li}_3(1-d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.83377, size = 280, normalized size = 1.28 \[ \frac{2 \left (2 x \text{PolyLog}(2,1-d x) \left (\left (a d^2-c\right ) \log (1-d x)+a d^2\right )-a d^2 x \text{PolyLog}(3,d x)-2 a d^2 x \text{PolyLog}(3,1-d x)+2 c x \text{PolyLog}(3,1-d x)-a d^2 x \log ^2(1-d x)+a d^2 x \log (d x) \log ^2(1-d x)+2 a d^2 x \log (d x) \log (1-d x)+a d \log ^2(1-d x)+3 c d x^2+c d x^2 \log ^2(1-d x)-3 c d x^2 \log (1-d x)-c x \log ^2(1-d x)-c x \log (d x) \log ^2(1-d x)+3 c x \log (1-d x)-2 c x\right )+2 \text{PolyLog}(2,d x) \left ((d x-1) \log (1-d x) (a d+c x)-c d x^2\right )-b d x \text{PolyLog}(2,d x)^2}{2 d x} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x])/x^2,x]

[Out]

(2*(-(c*d*x^2) + (a*d + c*x)*(-1 + d*x)*Log[1 - d*x])*PolyLog[2, d*x] - b*d*x*PolyLog[2, d*x]^2 + 2*(-2*c*x +
3*c*d*x^2 + 3*c*x*Log[1 - d*x] - 3*c*d*x^2*Log[1 - d*x] + 2*a*d^2*x*Log[d*x]*Log[1 - d*x] + a*d*Log[1 - d*x]^2
- c*x*Log[1 - d*x]^2 - a*d^2*x*Log[1 - d*x]^2 + c*d*x^2*Log[1 - d*x]^2 - c*x*Log[d*x]*Log[1 - d*x]^2 + a*d^2*
x*Log[d*x]*Log[1 - d*x]^2 + 2*x*(a*d^2 + (-c + a*d^2)*Log[1 - d*x])*PolyLog[2, 1 - d*x] - a*d^2*x*PolyLog[3, d
*x] + 2*c*x*PolyLog[3, 1 - d*x] - 2*a*d^2*x*PolyLog[3, 1 - d*x]))/(2*d*x)

________________________________________________________________________________________

Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+bx+a \right ) \ln \left ( -dx+1 \right ){\it polylog} \left ( 2,dx \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)*ln(-d*x+1)*polylog(2,d*x)/x^2,x)

[Out]

int((c*x^2+b*x+a)*ln(-d*x+1)*polylog(2,d*x)/x^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x)/x^2,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)*dilog(d*x)*log(-d*x + 1)/x^2, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x)/x^2,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)*dilog(d*x)*log(-d*x + 1)/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)*ln(-d*x+1)*polylog(2,d*x)/x**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x)/x^2,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)*dilog(d*x)*log(-d*x + 1)/x^2, x)