3.138 \(\int (d+e x)^2 \text{PolyLog}(2,c (a+b x)) \, dx\)

Optimal. Leaf size=385 \[ -\frac{(b d-a e)^3 \text{PolyLog}(2,c (a+b x))}{3 b^3 e}+\frac{(d+e x)^3 \text{PolyLog}(2,c (a+b x))}{3 e}-\frac{x (-a c e+b c d+e)^2}{9 b^2 c^2}-\frac{(b d-a e) (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{6 b^3 c^2 e}-\frac{(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{9 b^3 c^3 e}-\frac{x (b d-a e) (-a c e+b c d+e)}{6 b^2 c}-\frac{(-a c-b c x+1) (b d-a e)^2 \log (-a c-b c x+1)}{3 b^3 c}-\frac{x (b d-a e)^2}{3 b^2}-\frac{(d+e x)^2 (-a c e+b c d+e)}{18 b c e}+\frac{(d+e x)^2 (b d-a e) \log (-a c-b c x+1)}{6 b e}+\frac{(d+e x)^3 \log (-a c-b c x+1)}{9 e}-\frac{(d+e x)^2 (b d-a e)}{12 b e}-\frac{(d+e x)^3}{27 e} \]

[Out]

-((b*d - a*e)^2*x)/(3*b^2) - ((b*d - a*e)*(b*c*d + e - a*c*e)*x)/(6*b^2*c) - ((b*c*d + e - a*c*e)^2*x)/(9*b^2*
c^2) - ((b*d - a*e)*(d + e*x)^2)/(12*b*e) - ((b*c*d + e - a*c*e)*(d + e*x)^2)/(18*b*c*e) - (d + e*x)^3/(27*e)
- ((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) - ((b*c*d + e - a*c*e)^3*Log[1 - a*c
- b*c*x])/(9*b^3*c^3*e) - ((b*d - a*e)^2*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(3*b^3*c) + ((b*d - a*e)*(d +
e*x)^2*Log[1 - a*c - b*c*x])/(6*b*e) + ((d + e*x)^3*Log[1 - a*c - b*c*x])/(9*e) - ((b*d - a*e)^3*PolyLog[2, c
*(a + b*x)])/(3*b^3*e) + ((d + e*x)^3*PolyLog[2, c*(a + b*x)])/(3*e)

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Rubi [A]  time = 0.338798, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {6598, 2418, 2389, 2295, 2393, 2391, 2395, 43} \[ -\frac{(b d-a e)^3 \text{PolyLog}(2,c (a+b x))}{3 b^3 e}+\frac{(d+e x)^3 \text{PolyLog}(2,c (a+b x))}{3 e}-\frac{x (-a c e+b c d+e)^2}{9 b^2 c^2}-\frac{(b d-a e) (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{6 b^3 c^2 e}-\frac{(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{9 b^3 c^3 e}-\frac{x (b d-a e) (-a c e+b c d+e)}{6 b^2 c}-\frac{(-a c-b c x+1) (b d-a e)^2 \log (-a c-b c x+1)}{3 b^3 c}-\frac{x (b d-a e)^2}{3 b^2}-\frac{(d+e x)^2 (-a c e+b c d+e)}{18 b c e}+\frac{(d+e x)^2 (b d-a e) \log (-a c-b c x+1)}{6 b e}+\frac{(d+e x)^3 \log (-a c-b c x+1)}{9 e}-\frac{(d+e x)^2 (b d-a e)}{12 b e}-\frac{(d+e x)^3}{27 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*PolyLog[2, c*(a + b*x)],x]

[Out]

-((b*d - a*e)^2*x)/(3*b^2) - ((b*d - a*e)*(b*c*d + e - a*c*e)*x)/(6*b^2*c) - ((b*c*d + e - a*c*e)^2*x)/(9*b^2*
c^2) - ((b*d - a*e)*(d + e*x)^2)/(12*b*e) - ((b*c*d + e - a*c*e)*(d + e*x)^2)/(18*b*c*e) - (d + e*x)^3/(27*e)
- ((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) - ((b*c*d + e - a*c*e)^3*Log[1 - a*c
- b*c*x])/(9*b^3*c^3*e) - ((b*d - a*e)^2*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(3*b^3*c) + ((b*d - a*e)*(d +
e*x)^2*Log[1 - a*c - b*c*x])/(6*b*e) + ((d + e*x)^3*Log[1 - a*c - b*c*x])/(9*e) - ((b*d - a*e)^3*PolyLog[2, c
*(a + b*x)])/(3*b^3*e) + ((d + e*x)^3*PolyLog[2, c*(a + b*x)])/(3*e)

Rule 6598

Int[((d_.) + (e_.)*(x_))^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> Simp[((d + e*x)^(m + 1)*Po
lyLog[2, c*(a + b*x)])/(e*(m + 1)), x] + Dist[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]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> 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] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d+e x)^2 \text{Li}_2(c (a+b x)) \, dx &=\frac{(d+e x)^3 \text{Li}_2(c (a+b x))}{3 e}+\frac{b \int \frac{(d+e x)^3 \log (1-a c-b c x)}{a+b x} \, dx}{3 e}\\ &=\frac{(d+e x)^3 \text{Li}_2(c (a+b x))}{3 e}+\frac{b \int \left (\frac{e (b d-a e)^2 \log (1-a c-b c x)}{b^3}+\frac{(b d-a e)^3 \log (1-a c-b c x)}{b^3 (a+b x)}+\frac{e (b d-a e) (d+e x) \log (1-a c-b c x)}{b^2}+\frac{e (d+e x)^2 \log (1-a c-b c x)}{b}\right ) \, dx}{3 e}\\ &=\frac{(d+e x)^3 \text{Li}_2(c (a+b x))}{3 e}+\frac{1}{3} \int (d+e x)^2 \log (1-a c-b c x) \, dx+\frac{(b d-a e) \int (d+e x) \log (1-a c-b c x) \, dx}{3 b}+\frac{(b d-a e)^2 \int \log (1-a c-b c x) \, dx}{3 b^2}+\frac{(b d-a e)^3 \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{3 b^2 e}\\ &=\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{(d+e x)^3 \text{Li}_2(c (a+b x))}{3 e}+\frac{(b c) \int \frac{(d+e x)^3}{1-a c-b c x} \, dx}{9 e}+\frac{(c (b d-a e)) \int \frac{(d+e x)^2}{1-a c-b c x} \, dx}{6 e}-\frac{(b d-a e)^2 \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b^3 c}+\frac{(b d-a e)^3 \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 b^3 e}\\ &=-\frac{(b d-a e)^2 x}{3 b^2}-\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 \text{Li}_2(c (a+b x))}{3 b^3 e}+\frac{(d+e x)^3 \text{Li}_2(c (a+b x))}{3 e}+\frac{(b c) \int \left (-\frac{e (b c d+e-a c e)^2}{b^3 c^3}+\frac{(b c d+e-a c e)^3}{b^3 c^3 (1-a c-b c x)}-\frac{e (b c d+e-a c e) (d+e x)}{b^2 c^2}-\frac{e (d+e x)^2}{b c}\right ) \, dx}{9 e}+\frac{(c (b d-a e)) \int \left (-\frac{e (b c d+e-a c e)}{b^2 c^2}+\frac{(b c d+e-a c e)^2}{b^2 c^2 (1-a c-b c x)}-\frac{e (d+e x)}{b c}\right ) \, dx}{6 e}\\ &=-\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 \text{Li}_2(c (a+b x))}{3 b^3 e}+\frac{(d+e x)^3 \text{Li}_2(c (a+b x))}{3 e}\\ \end{align*}

Mathematica [A]  time = 0.182727, size = 274, normalized size = 0.71 \[ \frac{36 c^3 \left (-3 a^2 b d e+a^3 e^2+3 a b^2 d^2+b^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \text{PolyLog}(2,c (a+b x))+b c \left (-66 a^2 c^2 e^2 x+3 a c \left (b c \left (-36 d^2+54 d e x+5 e^2 x^2\right )+14 e^2 x\right )+108 b c d^2 (a c+b c x-1) \log (1-c (a+b x))-x \left (b^2 c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+6 b c e (9 d+e x)+12 e^2\right )\right )+6 e (a c+b c x-1) \log (-a c-b c x+1) \left (e \left (11 a^2 c^2-7 a c+2\right )+b c (d (9-27 a c)+e x (2-5 a c))+b^2 c^2 x (9 d+2 e x)\right )}{108 b^3 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*PolyLog[2, c*(a + b*x)],x]

[Out]

(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 +
27*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))*
PolyLog[2, c*(a + b*x)])/(108*b^3*c^3)

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Maple [A]  time = 0.058, size = 687, normalized size = 1.8 \begin{align*} -{d}^{2}x-{\frac{{x}^{3}{e}^{2}}{27}}-{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ){a}^{2}}{c{b}^{3}}}+{\frac{3\,axde}{2\,b}}+{\frac{7\,{e}^{2}xa}{18\,{b}^{2}c}}-{\frac{e{\it dilog} \left ( -xbc-ac+1 \right ){a}^{2}d}{{b}^{2}}}-{\frac{dxe}{2\,bc}}-{\frac{3\,e\ln \left ( -xbc-ac+1 \right ){a}^{2}d}{2\,{b}^{2}}}-{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ){x}^{2}a}{6\,b}}+{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ) x{a}^{2}}{3\,{b}^{2}}}-{\frac{e\ln \left ( -xbc-ac+1 \right ) d}{2\,{b}^{2}{c}^{2}}}-{\frac{5\,aed}{2\,{b}^{2}c}}+{\frac{7\,e{a}^{2}d}{4\,{b}^{2}}}+{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ) a}{2\,{b}^{3}{c}^{2}}}+{\frac{{d}^{2}}{bc}}+{\frac{11\,{e}^{2}}{54\,{c}^{3}{b}^{3}}}+{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ){x}^{3}}{9}}-{\frac{{\it dilog} \left ( -xbc-ac+1 \right ){d}^{3}}{3\,e}}+{\frac{{\it polylog} \left ( 2,xbc+ac \right ){d}^{3}}{3\,e}}+\ln \left ( -xbc-ac+1 \right ) x{d}^{2}+{\frac{{e}^{2}{\it polylog} \left ( 2,xbc+ac \right ){x}^{3}}{3}}+{\it polylog} \left ( 2,xbc+ac \right ) x{d}^{2}-{\frac{{x}^{2}de}{4}}+2\,{\frac{e\ln \left ( -xbc-ac+1 \right ) ad}{{b}^{2}c}}-{\frac{e\ln \left ( -xbc-ac+1 \right ) xad}{b}}+{\frac{13\,{e}^{2}{a}^{2}}{9\,c{b}^{3}}}+{\frac{3\,de}{4\,{b}^{2}{c}^{2}}}-{\frac{31\,a{e}^{2}}{36\,{b}^{3}{c}^{2}}}-{\frac{85\,{e}^{2}{a}^{3}}{108\,{b}^{3}}}-{\frac{a{d}^{2}}{b}}+{\frac{5\,{e}^{2}{x}^{2}a}{36\,b}}-{\frac{11\,{e}^{2}x{a}^{2}}{18\,{b}^{2}}}-{\frac{{e}^{2}{x}^{2}}{18\,bc}}-{\frac{{e}^{2}x}{9\,{b}^{2}{c}^{2}}}+{\frac{e\ln \left ( -xbc-ac+1 \right ){x}^{2}d}{2}}+e{\it polylog} \left ( 2,xbc+ac \right ) d{x}^{2}+{\frac{{e}^{2}{\it dilog} \left ( -xbc-ac+1 \right ){a}^{3}}{3\,{b}^{3}}}+{\frac{11\,{e}^{2}\ln \left ( -xbc-ac+1 \right ){a}^{3}}{18\,{b}^{3}}}+{\frac{{\it dilog} \left ( -xbc-ac+1 \right ) a{d}^{2}}{b}}+{\frac{\ln \left ( -xbc-ac+1 \right ) a{d}^{2}}{b}}-{\frac{\ln \left ( -xbc-ac+1 \right ){d}^{2}}{bc}}-{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ) }{9\,{c}^{3}{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*polylog(2,c*(b*x+a)),x)

[Out]

-d^2*x-1/27*x^3*e^2-1/b^3/c*e^2*ln(-b*c*x-a*c+1)*a^2+3/2/b*e*x*a*d+7/18/b^2/c*e^2*x*a-1/b^2*e*dilog(-b*c*x-a*c
+1)*a^2*d-1/2/b/c*x*d*e-3/2/b^2*e*ln(-b*c*x-a*c+1)*a^2*d-1/6/b*e^2*ln(-b*c*x-a*c+1)*x^2*a+1/3/b^2*e^2*ln(-b*c*
x-a*c+1)*x*a^2-1/2/b^2/c^2*e*ln(-b*c*x-a*c+1)*d-5/2/b^2/c*e*a*d+7/4/b^2*e*a^2*d+1/2/b^3/c^2*e^2*ln(-b*c*x-a*c+
1)*a+1/b/c*d^2+11/54/b^3/c^3*e^2+1/9*e^2*ln(-b*c*x-a*c+1)*x^3-1/3/e*dilog(-b*c*x-a*c+1)*d^3+1/3/e*polylog(2,b*
c*x+a*c)*d^3+ln(-b*c*x-a*c+1)*x*d^2+1/3*e^2*polylog(2,b*c*x+a*c)*x^3+polylog(2,b*c*x+a*c)*x*d^2-1/4*x^2*d*e+2/
b^2/c*e*ln(-b*c*x-a*c+1)*a*d-1/b*e*ln(-b*c*x-a*c+1)*x*a*d+13/9/b^3/c*e^2*a^2+3/4/b^2/c^2*e*d-31/36/b^3/c^2*e^2
*a-85/108/b^3*e^2*a^3-1/b*a*d^2+5/36/b*e^2*x^2*a-11/18/b^2*e^2*x*a^2-1/18/b/c*x^2*e^2-1/9/b^2/c^2*e^2*x+1/2*e*
ln(-b*c*x-a*c+1)*x^2*d+e*polylog(2,b*c*x+a*c)*d*x^2+1/3/b^3*e^2*dilog(-b*c*x-a*c+1)*a^3+11/18/b^3*e^2*ln(-b*c*
x-a*c+1)*a^3+1/b*dilog(-b*c*x-a*c+1)*a*d^2+1/b*ln(-b*c*x-a*c+1)*a*d^2-1/b/c*ln(-b*c*x-a*c+1)*d^2-1/9/b^3/c^3*e
^2*ln(-b*c*x-a*c+1)

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Maxima [A]  time = 1.03377, size = 548, normalized size = 1.42 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*polylog(2,c*(b*x+a)),x, algorithm="maxima")

[Out]

-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 - 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)*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)

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Fricas [A]  time = 2.32108, size = 779, normalized size = 2.02 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*polylog(2,c*(b*x+a)),x, algorithm="fricas")

[Out]

-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)

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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((e*x+d)**2*polylog(2,c*(b*x+a)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*polylog(2,c*(b*x+a)),x, algorithm="giac")

[Out]

integrate((e*x + d)^2*dilog((b*x + a)*c), x)