3.2.37 \(\int (d+e x)^3 \text {PolyLog}(2,c (a+b x)) \, dx\) [137]

Optimal. Leaf size=605 \[ -\frac {(b d-a e)^3 x}{4 b^3}-\frac {(b d-a e)^2 (b c d+e-a c e) x}{8 b^3 c}-\frac {(b d-a e) (b c d+e-a c e)^2 x}{12 b^3 c^2}-\frac {(b c d+e-a c e)^3 x}{16 b^3 c^3}-\frac {(b d-a e)^2 (d+e x)^2}{16 b^2 e}-\frac {(b d-a e) (b c d+e-a c e) (d+e x)^2}{24 b^2 c e}-\frac {(b c d+e-a c e)^2 (d+e x)^2}{32 b^2 c^2 e}-\frac {(b d-a e) (d+e x)^3}{36 b e}-\frac {(b c d+e-a c e) (d+e x)^3}{48 b c e}-\frac {(d+e x)^4}{64 e}-\frac {(b d-a e)^2 (b c d+e-a c e)^2 \log (1-a c-b c x)}{8 b^4 c^2 e}-\frac {(b d-a e) (b c d+e-a c e)^3 \log (1-a c-b c x)}{12 b^4 c^3 e}-\frac {(b c d+e-a c e)^4 \log (1-a c-b c x)}{16 b^4 c^4 e}-\frac {(b d-a e)^3 (1-a c-b c x) \log (1-a c-b c x)}{4 b^4 c}+\frac {(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac {(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac {(d+e x)^4 \log (1-a c-b c x)}{16 e}-\frac {(b d-a e)^4 \text {PolyLog}(2,c (a+b x))}{4 b^4 e}+\frac {(d+e x)^4 \text {PolyLog}(2,c (a+b x))}{4 e} \]

[Out]

-1/4*(-a*e+b*d)^3*x/b^3-1/8*(-a*e+b*d)^2*(-a*c*e+b*c*d+e)*x/b^3/c-1/12*(-a*e+b*d)*(-a*c*e+b*c*d+e)^2*x/b^3/c^2
-1/16*(-a*c*e+b*c*d+e)^3*x/b^3/c^3-1/16*(-a*e+b*d)^2*(e*x+d)^2/b^2/e-1/24*(-a*e+b*d)*(-a*c*e+b*c*d+e)*(e*x+d)^
2/b^2/c/e-1/32*(-a*c*e+b*c*d+e)^2*(e*x+d)^2/b^2/c^2/e-1/36*(-a*e+b*d)*(e*x+d)^3/b/e-1/48*(-a*c*e+b*c*d+e)*(e*x
+d)^3/b/c/e-1/64*(e*x+d)^4/e-1/8*(-a*e+b*d)^2*(-a*c*e+b*c*d+e)^2*ln(-b*c*x-a*c+1)/b^4/c^2/e-1/12*(-a*e+b*d)*(-
a*c*e+b*c*d+e)^3*ln(-b*c*x-a*c+1)/b^4/c^3/e-1/16*(-a*c*e+b*c*d+e)^4*ln(-b*c*x-a*c+1)/b^4/c^4/e-1/4*(-a*e+b*d)^
3*(-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)/b^4/c+1/8*(-a*e+b*d)^2*(e*x+d)^2*ln(-b*c*x-a*c+1)/b^2/e+1/12*(-a*e+b*d)*(e*x
+d)^3*ln(-b*c*x-a*c+1)/b/e+1/16*(e*x+d)^4*ln(-b*c*x-a*c+1)/e-1/4*(-a*e+b*d)^4*polylog(2,c*(b*x+a))/b^4/e+1/4*(
e*x+d)^4*polylog(2,c*(b*x+a))/e

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Rubi [A]
time = 0.42, antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6733, 2465, 2436, 2332, 2440, 2438, 2442, 45} \begin {gather*} -\frac {(-a c e+b c d+e)^4 \log (-a c-b c x+1)}{16 b^4 c^4 e}-\frac {(b d-a e) (-a c e+b c d+e)^3 \log (-a c-b c x+1)}{12 b^4 c^3 e}-\frac {(b d-a e)^2 (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{8 b^4 c^2 e}-\frac {(b d-a e)^4 \text {Li}_2(c (a+b x))}{4 b^4 e}-\frac {(-a c-b c x+1) (b d-a e)^3 \log (-a c-b c x+1)}{4 b^4 c}-\frac {x (-a c e+b c d+e)^3}{16 b^3 c^3}-\frac {x (b d-a e) (-a c e+b c d+e)^2}{12 b^3 c^2}-\frac {x (b d-a e)^2 (-a c e+b c d+e)}{8 b^3 c}-\frac {x (b d-a e)^3}{4 b^3}-\frac {(d+e x)^2 (-a c e+b c d+e)^2}{32 b^2 c^2 e}-\frac {(d+e x)^2 (b d-a e) (-a c e+b c d+e)}{24 b^2 c e}+\frac {(d+e x)^2 (b d-a e)^2 \log (-a c-b c x+1)}{8 b^2 e}-\frac {(d+e x)^2 (b d-a e)^2}{16 b^2 e}+\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}-\frac {(d+e x)^3 (-a c e+b c d+e)}{48 b c e}+\frac {(d+e x)^3 (b d-a e) \log (-a c-b c x+1)}{12 b e}+\frac {(d+e x)^4 \log (-a c-b c x+1)}{16 e}-\frac {(d+e x)^3 (b d-a e)}{36 b e}-\frac {(d+e x)^4}{64 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*PolyLog[2, c*(a + b*x)],x]

[Out]

-1/4*((b*d - a*e)^3*x)/b^3 - ((b*d - a*e)^2*(b*c*d + e - a*c*e)*x)/(8*b^3*c) - ((b*d - a*e)*(b*c*d + e - a*c*e
)^2*x)/(12*b^3*c^2) - ((b*c*d + e - a*c*e)^3*x)/(16*b^3*c^3) - ((b*d - a*e)^2*(d + e*x)^2)/(16*b^2*e) - ((b*d
- a*e)*(b*c*d + e - a*c*e)*(d + e*x)^2)/(24*b^2*c*e) - ((b*c*d + e - a*c*e)^2*(d + e*x)^2)/(32*b^2*c^2*e) - ((
b*d - a*e)*(d + e*x)^3)/(36*b*e) - ((b*c*d + e - a*c*e)*(d + e*x)^3)/(48*b*c*e) - (d + e*x)^4/(64*e) - ((b*d -
a*e)^2*(b*c*d + e - a*c*e)^2*Log[1 - a*c - b*c*x])/(8*b^4*c^2*e) - ((b*d - a*e)*(b*c*d + e - a*c*e)^3*Log[1 -
a*c - b*c*x])/(12*b^4*c^3*e) - ((b*c*d + e - a*c*e)^4*Log[1 - a*c - b*c*x])/(16*b^4*c^4*e) - ((b*d - a*e)^3*(
1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(4*b^4*c) + ((b*d - a*e)^2*(d + e*x)^2*Log[1 - a*c - b*c*x])/(8*b^2*e)
+ ((b*d - a*e)*(d + e*x)^3*Log[1 - a*c - b*c*x])/(12*b*e) + ((d + e*x)^4*Log[1 - a*c - b*c*x])/(16*e) - ((b*d
- a*e)^4*PolyLog[2, c*(a + b*x)])/(4*b^4*e) + ((d + e*x)^4*PolyLog[2, c*(a + b*x)])/(4*e)

Rule 45

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])

Rule 2332

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

Rule 2436

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 2438

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 2440

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 2442

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 2465

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 6733

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]

Rubi steps

\begin {align*} \int (d+e x)^3 \text {Li}_2(c (a+b x)) \, dx &=\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}+\frac {b \int \frac {(d+e x)^4 \log (1-a c-b c x)}{a+b x} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}+\frac {b \int \left (\frac {e (b d-a e)^3 \log (1-a c-b c x)}{b^4}+\frac {(b d-a e)^4 \log (1-a c-b c x)}{b^4 (a+b x)}+\frac {e (b d-a e)^2 (d+e x) \log (1-a c-b c x)}{b^3}+\frac {e (b d-a e) (d+e x)^2 \log (1-a c-b c x)}{b^2}+\frac {e (d+e x)^3 \log (1-a c-b c x)}{b}\right ) \, dx}{4 e}\\ &=\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}+\frac {1}{4} \int (d+e x)^3 \log (1-a c-b c x) \, dx+\frac {(b d-a e) \int (d+e x)^2 \log (1-a c-b c x) \, dx}{4 b}+\frac {(b d-a e)^2 \int (d+e x) \log (1-a c-b c x) \, dx}{4 b^2}+\frac {(b d-a e)^3 \int \log (1-a c-b c x) \, dx}{4 b^3}+\frac {(b d-a e)^4 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{4 b^3 e}\\ &=\frac {(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac {(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac {(d+e x)^4 \log (1-a c-b c x)}{16 e}+\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}+\frac {(b c) \int \frac {(d+e x)^4}{1-a c-b c x} \, dx}{16 e}+\frac {(c (b d-a e)) \int \frac {(d+e x)^3}{1-a c-b c x} \, dx}{12 e}+\frac {\left (c (b d-a e)^2\right ) \int \frac {(d+e x)^2}{1-a c-b c x} \, dx}{8 b e}-\frac {(b d-a e)^3 \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{4 b^4 c}+\frac {(b d-a e)^4 \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{4 b^4 e}\\ &=-\frac {(b d-a e)^3 x}{4 b^3}-\frac {(b d-a e)^3 (1-a c-b c x) \log (1-a c-b c x)}{4 b^4 c}+\frac {(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac {(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac {(d+e x)^4 \log (1-a c-b c x)}{16 e}-\frac {(b d-a e)^4 \text {Li}_2(c (a+b x))}{4 b^4 e}+\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}+\frac {(b c) \int \left (-\frac {e (b c d+e-a c e)^3}{b^4 c^4}+\frac {(b c d+e-a c e)^4}{b^4 c^4 (1-a c-b c x)}-\frac {e (b c d+e-a c e)^2 (d+e x)}{b^3 c^3}-\frac {e (b c d+e-a c e) (d+e x)^2}{b^2 c^2}-\frac {e (d+e x)^3}{b c}\right ) \, dx}{16 e}+\frac {(c (b d-a e)) \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}{12 e}+\frac {\left (c (b d-a e)^2\right ) \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}{8 b e}\\ &=-\frac {(b d-a e)^3 x}{4 b^3}-\frac {(b d-a e)^2 (b c d+e-a c e) x}{8 b^3 c}-\frac {(b d-a e) (b c d+e-a c e)^2 x}{12 b^3 c^2}-\frac {(b c d+e-a c e)^3 x}{16 b^3 c^3}-\frac {(b d-a e)^2 (d+e x)^2}{16 b^2 e}-\frac {(b d-a e) (b c d+e-a c e) (d+e x)^2}{24 b^2 c e}-\frac {(b c d+e-a c e)^2 (d+e x)^2}{32 b^2 c^2 e}-\frac {(b d-a e) (d+e x)^3}{36 b e}-\frac {(b c d+e-a c e) (d+e x)^3}{48 b c e}-\frac {(d+e x)^4}{64 e}-\frac {(b d-a e)^2 (b c d+e-a c e)^2 \log (1-a c-b c x)}{8 b^4 c^2 e}-\frac {(b d-a e) (b c d+e-a c e)^3 \log (1-a c-b c x)}{12 b^4 c^3 e}-\frac {(b c d+e-a c e)^4 \log (1-a c-b c x)}{16 b^4 c^4 e}-\frac {(b d-a e)^3 (1-a c-b c x) \log (1-a c-b c x)}{4 b^4 c}+\frac {(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac {(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac {(d+e x)^4 \log (1-a c-b c x)}{16 e}-\frac {(b d-a e)^4 \text {Li}_2(c (a+b x))}{4 b^4 e}+\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 485, normalized size = 0.80 \begin {gather*} \frac {12 e (-1+a c+b c x) \left (\left (3-13 a c+23 a^2 c^2-25 a^3 c^3\right ) e^2+b c e \left (8 \left (2-7 a c+11 a^2 c^2\right ) d+\left (3-10 a c+13 a^2 c^2\right ) e x\right )+b^3 c^3 x \left (36 d^2+16 d e x+3 e^2 x^2\right )+b^2 c^2 \left (-36 (-1+3 a c) d^2-8 (-2+5 a c) d e x+(3-7 a c) e^2 x^2\right )\right ) \log (1-a c-b c x)+b c \left (300 a^3 c^3 e^3 x-6 a^2 c^2 e^2 x (46 e+b c (176 d+13 e x))+4 a c \left (39 e^3 x+3 b c e^2 x (56 d+5 e x)+b^2 c^2 \left (-144 d^3+324 d^2 e x+60 d e^2 x^2+7 e^3 x^3\right )\right )-x \left (36 e^3+6 b c e^2 (32 d+3 e x)+12 b^2 c^2 e \left (36 d^2+8 d e x+e^2 x^2\right )+b^3 c^3 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )+576 b^2 c^2 d^3 (-1+a c+b c x) \log (1-c (a+b x))\right )-144 c^4 \left (-4 a b^3 d^3+6 a^2 b^2 d^2 e-4 a^3 b d e^2+a^4 e^3-b^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {PolyLog}(2,c (a+b x))}{576 b^4 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*PolyLog[2, c*(a + b*x)],x]

[Out]

(12*e*(-1 + a*c + b*c*x)*((3 - 13*a*c + 23*a^2*c^2 - 25*a^3*c^3)*e^2 + b*c*e*(8*(2 - 7*a*c + 11*a^2*c^2)*d + (
3 - 10*a*c + 13*a^2*c^2)*e*x) + b^3*c^3*x*(36*d^2 + 16*d*e*x + 3*e^2*x^2) + b^2*c^2*(-36*(-1 + 3*a*c)*d^2 - 8*
(-2 + 5*a*c)*d*e*x + (3 - 7*a*c)*e^2*x^2))*Log[1 - a*c - b*c*x] + b*c*(300*a^3*c^3*e^3*x - 6*a^2*c^2*e^2*x*(46
*e + b*c*(176*d + 13*e*x)) + 4*a*c*(39*e^3*x + 3*b*c*e^2*x*(56*d + 5*e*x) + b^2*c^2*(-144*d^3 + 324*d^2*e*x +
60*d*e^2*x^2 + 7*e^3*x^3)) - x*(36*e^3 + 6*b*c*e^2*(32*d + 3*e*x) + 12*b^2*c^2*e*(36*d^2 + 8*d*e*x + e^2*x^2)
+ b^3*c^3*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3)) + 576*b^2*c^2*d^3*(-1 + a*c + b*c*x)*Log[1 - c*(
a + b*x)]) - 144*c^4*(-4*a*b^3*d^3 + 6*a^2*b^2*d^2*e - 4*a^3*b*d*e^2 + a^4*e^3 - b^4*x*(4*d^3 + 6*d^2*e*x + 4*
d*e^2*x^2 + e^3*x^3))*PolyLog[2, c*(a + b*x)])/(576*b^4*c^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1145\) vs. \(2(567)=1134\).
time = 0.80, size = 1146, normalized size = 1.89 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*polylog(2,c*(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/c*(1/4*c/b^3*e^3*polylog(2,b*c*x+a*c)*a^4-c/b^2*e^2*polylog(2,b*c*x+a*c)*a^3*d+3/2*c/b*e*polylog(2,b*c*x+a
*c)*a^2*d^2-c*polylog(2,b*c*x+a*c)*a*d^3+1/4*c*b/e*polylog(2,b*c*x+a*c)*d^4-1/b^3*e^3*polylog(2,b*c*x+a*c)*a^3
*(b*c*x+a*c)+3/b^2*e^2*polylog(2,b*c*x+a*c)*a^2*d*(b*c*x+a*c)-3/b*e*polylog(2,b*c*x+a*c)*a*d^2*(b*c*x+a*c)+pol
ylog(2,b*c*x+a*c)*d^3*(b*c*x+a*c)+3/2/c/b^3*e^3*polylog(2,b*c*x+a*c)*a^2*(b*c*x+a*c)^2-3/c/b^2*e^2*polylog(2,b
*c*x+a*c)*a*d*(b*c*x+a*c)^2+3/2/c/b*e*polylog(2,b*c*x+a*c)*d^2*(b*c*x+a*c)^2-1/c^2/b^3*e^3*polylog(2,b*c*x+a*c
)*a*(b*c*x+a*c)^3+1/c^2/b^2*e^2*polylog(2,b*c*x+a*c)*d*(b*c*x+a*c)^3+1/4/c^3/b^3*e^3*polylog(2,b*c*x+a*c)*(b*c
*x+a*c)^4+1/4/c^3/b^3/e*((1/4*(-b*c*x-a*c+1)^4*ln(-b*c*x-a*c+1)-1/16*(-b*c*x-a*c+1)^4)*e^4-(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)*(-4*a*c*e^4+4*b*c*d*e^3+3*e^4)-(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)*(-6*a^2*c^2*e^4+12*a*b*c^2*d*e^3-6*b^2*c^2*d^2*e^2+8*a*c*e^4-8*b*c*d*e^3-3*e^4)+4*((-b
*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a^3*c^3*e^4-12*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a^2*b*c
^3*d*e^3+12*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a*b^2*c^3*d^2*e^2-4*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)
-1+x*b*c+a*c)*b^3*c^3*d^3*e-6*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a^2*c^2*e^4+12*((-b*c*x-a*c+1)*ln(
-b*c*x-a*c+1)-1+x*b*c+a*c)*a*b*c^2*d*e^3-6*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*b^2*c^2*d^2*e^2+4*((-
b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a*c*e^4-4*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*b*c*d*e^3-e
^4*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)-dilog(-b*c*x-a*c+1)*c^4*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*
e^2-4*a*b^3*d^3*e+b^4*d^4)))

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Maxima [A]
time = 0.28, size = 696, normalized size = 1.15 \begin {gather*} -\frac {{\left (4 \, a b^{3} d^{3} - 6 \, a^{2} b^{2} d^{2} e + 4 \, a^{3} b d e^{2} - a^{4} e^{3}\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 )}}{4 \, b^{4}} - \frac {9 \, b^{4} c^{4} x^{4} e^{3} + 4 \, {\left (16 \, b^{4} c^{4} d e^{2} - 7 \, a b^{3} c^{4} e^{3} + 3 \, b^{3} c^{3} e^{3}\right )} x^{3} + 6 \, {\left (36 \, b^{4} c^{4} d^{2} e + 13 \, a^{2} b^{2} c^{4} e^{3} - 10 \, a b^{2} c^{3} e^{3} + 3 \, b^{2} c^{2} e^{3} - 8 \, {\left (5 \, a b^{3} c^{4} e^{2} - 2 \, b^{3} c^{3} e^{2}\right )} d\right )} x^{2} + 12 \, {\left (48 \, b^{4} c^{4} d^{3} - 25 \, a^{3} b c^{4} e^{3} + 23 \, a^{2} b c^{3} e^{3} - 13 \, a b c^{2} e^{3} - 36 \, {\left (3 \, a b^{3} c^{4} e - b^{3} c^{3} e\right )} d^{2} + 3 \, b c e^{3} + 8 \, {\left (11 \, a^{2} b^{2} c^{4} e^{2} - 7 \, a b^{2} c^{3} e^{2} + 2 \, b^{2} c^{2} e^{2}\right )} d\right )} x - 144 \, {\left (b^{4} c^{4} x^{4} e^{3} + 4 \, b^{4} c^{4} d x^{3} e^{2} + 6 \, b^{4} c^{4} d^{2} x^{2} e + 4 \, b^{4} c^{4} d^{3} x\right )} {\rm Li}_2\left (b c x + a c\right ) - 12 \, {\left (3 \, b^{4} c^{4} x^{4} e^{3} - 25 \, a^{4} c^{4} e^{3} + 48 \, a^{3} c^{3} e^{3} - 36 \, a^{2} c^{2} e^{3} + 48 \, {\left (a b^{3} c^{4} - b^{3} c^{3}\right )} d^{3} + 4 \, {\left (4 \, b^{4} c^{4} d e^{2} - a b^{3} c^{4} e^{3}\right )} x^{3} - 36 \, {\left (3 \, a^{2} b^{2} c^{4} e - 4 \, a b^{2} c^{3} e + b^{2} c^{2} e\right )} d^{2} + 6 \, {\left (6 \, b^{4} c^{4} d^{2} e - 4 \, a b^{3} c^{4} d e^{2} + a^{2} b^{2} c^{4} e^{3}\right )} x^{2} + 16 \, a c e^{3} + 8 \, {\left (11 \, a^{3} b c^{4} e^{2} - 18 \, a^{2} b c^{3} e^{2} + 9 \, a b c^{2} e^{2} - 2 \, b c e^{2}\right )} d + 12 \, {\left (4 \, b^{4} c^{4} d^{3} - 6 \, a b^{3} c^{4} d^{2} e + 4 \, a^{2} b^{2} c^{4} d e^{2} - a^{3} b c^{4} e^{3}\right )} x - 3 \, e^{3}\right )} \log \left (-b c x - a c + 1\right )}{576 \, b^{4} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*polylog(2,c*(b*x+a)),x, algorithm="maxima")

[Out]

-1/4*(4*a*b^3*d^3 - 6*a^2*b^2*d^2*e + 4*a^3*b*d*e^2 - a^4*e^3)*(log(b*c*x + a*c)*log(-b*c*x - a*c + 1) + dilog
(-b*c*x - a*c + 1))/b^4 - 1/576*(9*b^4*c^4*x^4*e^3 + 4*(16*b^4*c^4*d*e^2 - 7*a*b^3*c^4*e^3 + 3*b^3*c^3*e^3)*x^
3 + 6*(36*b^4*c^4*d^2*e + 13*a^2*b^2*c^4*e^3 - 10*a*b^2*c^3*e^3 + 3*b^2*c^2*e^3 - 8*(5*a*b^3*c^4*e^2 - 2*b^3*c
^3*e^2)*d)*x^2 + 12*(48*b^4*c^4*d^3 - 25*a^3*b*c^4*e^3 + 23*a^2*b*c^3*e^3 - 13*a*b*c^2*e^3 - 36*(3*a*b^3*c^4*e
- b^3*c^3*e)*d^2 + 3*b*c*e^3 + 8*(11*a^2*b^2*c^4*e^2 - 7*a*b^2*c^3*e^2 + 2*b^2*c^2*e^2)*d)*x - 144*(b^4*c^4*x
^4*e^3 + 4*b^4*c^4*d*x^3*e^2 + 6*b^4*c^4*d^2*x^2*e + 4*b^4*c^4*d^3*x)*dilog(b*c*x + a*c) - 12*(3*b^4*c^4*x^4*e
^3 - 25*a^4*c^4*e^3 + 48*a^3*c^3*e^3 - 36*a^2*c^2*e^3 + 48*(a*b^3*c^4 - b^3*c^3)*d^3 + 4*(4*b^4*c^4*d*e^2 - a*
b^3*c^4*e^3)*x^3 - 36*(3*a^2*b^2*c^4*e - 4*a*b^2*c^3*e + b^2*c^2*e)*d^2 + 6*(6*b^4*c^4*d^2*e - 4*a*b^3*c^4*d*e
^2 + a^2*b^2*c^4*e^3)*x^2 + 16*a*c*e^3 + 8*(11*a^3*b*c^4*e^2 - 18*a^2*b*c^3*e^2 + 9*a*b*c^2*e^2 - 2*b*c*e^2)*d
+ 12*(4*b^4*c^4*d^3 - 6*a*b^3*c^4*d^2*e + 4*a^2*b^2*c^4*d*e^2 - a^3*b*c^4*e^3)*x - 3*e^3)*log(-b*c*x - a*c +
1))/(b^4*c^4)

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Fricas [A]
time = 0.36, size = 625, normalized size = 1.03 \begin {gather*} -\frac {576 \, b^{4} c^{4} d^{3} x - 144 \, {\left (4 \, b^{4} c^{4} d^{3} x + 4 \, a b^{3} c^{4} d^{3} + {\left (b^{4} c^{4} x^{4} - a^{4} c^{4}\right )} e^{3} + 4 \, {\left (b^{4} c^{4} d x^{3} + a^{3} b c^{4} d\right )} e^{2} + 6 \, {\left (b^{4} c^{4} d^{2} x^{2} - a^{2} b^{2} c^{4} d^{2}\right )} e\right )} {\rm Li}_2\left (b c x + a c\right ) + {\left (9 \, b^{4} c^{4} x^{4} - 4 \, {\left (7 \, a b^{3} c^{4} - 3 \, b^{3} c^{3}\right )} x^{3} + 6 \, {\left (13 \, a^{2} b^{2} c^{4} - 10 \, a b^{2} c^{3} + 3 \, b^{2} c^{2}\right )} x^{2} - 12 \, {\left (25 \, a^{3} b c^{4} - 23 \, a^{2} b c^{3} + 13 \, a b c^{2} - 3 \, b c\right )} x\right )} e^{3} + 16 \, {\left (4 \, b^{4} c^{4} d x^{3} - 3 \, {\left (5 \, a b^{3} c^{4} - 2 \, b^{3} c^{3}\right )} d x^{2} + 6 \, {\left (11 \, a^{2} b^{2} c^{4} - 7 \, a b^{2} c^{3} + 2 \, b^{2} c^{2}\right )} d x\right )} e^{2} + 216 \, {\left (b^{4} c^{4} d^{2} x^{2} - 2 \, {\left (3 \, a b^{3} c^{4} - b^{3} c^{3}\right )} d^{2} x\right )} e - 12 \, {\left (48 \, b^{4} c^{4} d^{3} x + 48 \, {\left (a b^{3} c^{4} - b^{3} c^{3}\right )} d^{3} + {\left (3 \, b^{4} c^{4} x^{4} - 4 \, a b^{3} c^{4} x^{3} + 6 \, a^{2} b^{2} c^{4} x^{2} - 12 \, a^{3} b c^{4} x - 25 \, a^{4} c^{4} + 48 \, a^{3} c^{3} - 36 \, a^{2} c^{2} + 16 \, a c - 3\right )} e^{3} + 8 \, {\left (2 \, b^{4} c^{4} d x^{3} - 3 \, a b^{3} c^{4} d x^{2} + 6 \, a^{2} b^{2} c^{4} d x + {\left (11 \, a^{3} b c^{4} - 18 \, a^{2} b c^{3} + 9 \, a b c^{2} - 2 \, b c\right )} d\right )} e^{2} + 36 \, {\left (b^{4} c^{4} d^{2} x^{2} - 2 \, a b^{3} c^{4} d^{2} x - {\left (3 \, a^{2} b^{2} c^{4} - 4 \, a b^{2} c^{3} + b^{2} c^{2}\right )} d^{2}\right )} e\right )} \log \left (-b c x - a c + 1\right )}{576 \, b^{4} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*polylog(2,c*(b*x+a)),x, algorithm="fricas")

[Out]

-1/576*(576*b^4*c^4*d^3*x - 144*(4*b^4*c^4*d^3*x + 4*a*b^3*c^4*d^3 + (b^4*c^4*x^4 - a^4*c^4)*e^3 + 4*(b^4*c^4*
d*x^3 + a^3*b*c^4*d)*e^2 + 6*(b^4*c^4*d^2*x^2 - a^2*b^2*c^4*d^2)*e)*dilog(b*c*x + a*c) + (9*b^4*c^4*x^4 - 4*(7
*a*b^3*c^4 - 3*b^3*c^3)*x^3 + 6*(13*a^2*b^2*c^4 - 10*a*b^2*c^3 + 3*b^2*c^2)*x^2 - 12*(25*a^3*b*c^4 - 23*a^2*b*
c^3 + 13*a*b*c^2 - 3*b*c)*x)*e^3 + 16*(4*b^4*c^4*d*x^3 - 3*(5*a*b^3*c^4 - 2*b^3*c^3)*d*x^2 + 6*(11*a^2*b^2*c^4
- 7*a*b^2*c^3 + 2*b^2*c^2)*d*x)*e^2 + 216*(b^4*c^4*d^2*x^2 - 2*(3*a*b^3*c^4 - b^3*c^3)*d^2*x)*e - 12*(48*b^4*
c^4*d^3*x + 48*(a*b^3*c^4 - b^3*c^3)*d^3 + (3*b^4*c^4*x^4 - 4*a*b^3*c^4*x^3 + 6*a^2*b^2*c^4*x^2 - 12*a^3*b*c^4
*x - 25*a^4*c^4 + 48*a^3*c^3 - 36*a^2*c^2 + 16*a*c - 3)*e^3 + 8*(2*b^4*c^4*d*x^3 - 3*a*b^3*c^4*d*x^2 + 6*a^2*b
^2*c^4*d*x + (11*a^3*b*c^4 - 18*a^2*b*c^3 + 9*a*b*c^2 - 2*b*c)*d)*e^2 + 36*(b^4*c^4*d^2*x^2 - 2*a*b^3*c^4*d^2*
x - (3*a^2*b^2*c^4 - 4*a*b^2*c^3 + b^2*c^2)*d^2)*e)*log(-b*c*x - a*c + 1))/(b^4*c^4)

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Sympy [A]
time = 20.82, size = 1030, normalized size = 1.70 \begin {gather*} \begin {cases} 0 & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\\left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) \operatorname {Li}_{2}\left (a c\right ) & \text {for}\: b = 0 \\\frac {25 a^{4} e^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{48 b^{4}} - \frac {a^{4} e^{3} \operatorname {Li}_{2}\left (a c + b c x\right )}{4 b^{4}} - \frac {11 a^{3} d e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{6 b^{3}} + \frac {a^{3} d e^{2} \operatorname {Li}_{2}\left (a c + b c x\right )}{b^{3}} + \frac {a^{3} e^{3} x \operatorname {Li}_{1}\left (a c + b c x\right )}{4 b^{3}} + \frac {25 a^{3} e^{3} x}{48 b^{3}} - \frac {a^{3} e^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{4} c} + \frac {9 a^{2} d^{2} e \operatorname {Li}_{1}\left (a c + b c x\right )}{4 b^{2}} - \frac {3 a^{2} d^{2} e \operatorname {Li}_{2}\left (a c + b c x\right )}{2 b^{2}} - \frac {a^{2} d e^{2} x \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{2}} - \frac {11 a^{2} d e^{2} x}{6 b^{2}} - \frac {a^{2} e^{3} x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{8 b^{2}} - \frac {13 a^{2} e^{3} x^{2}}{96 b^{2}} + \frac {3 a^{2} d e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{3} c} - \frac {23 a^{2} e^{3} x}{48 b^{3} c} + \frac {3 a^{2} e^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{4 b^{4} c^{2}} - \frac {a d^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{b} + \frac {a d^{3} \operatorname {Li}_{2}\left (a c + b c x\right )}{b} + \frac {3 a d^{2} e x \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b} + \frac {9 a d^{2} e x}{4 b} + \frac {a d e^{2} x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b} + \frac {5 a d e^{2} x^{2}}{12 b} + \frac {a e^{3} x^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{12 b} + \frac {7 a e^{3} x^{3}}{144 b} - \frac {3 a d^{2} e \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{2} c} + \frac {7 a d e^{2} x}{6 b^{2} c} + \frac {5 a e^{3} x^{2}}{48 b^{2} c} - \frac {3 a d e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{3} c^{2}} + \frac {13 a e^{3} x}{48 b^{3} c^{2}} - \frac {a e^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{3 b^{4} c^{3}} - d^{3} x \operatorname {Li}_{1}\left (a c + b c x\right ) + d^{3} x \operatorname {Li}_{2}\left (a c + b c x\right ) - d^{3} x - \frac {3 d^{2} e x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{4} + \frac {3 d^{2} e x^{2} \operatorname {Li}_{2}\left (a c + b c x\right )}{2} - \frac {3 d^{2} e x^{2}}{8} - \frac {d e^{2} x^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{3} + d e^{2} x^{3} \operatorname {Li}_{2}\left (a c + b c x\right ) - \frac {d e^{2} x^{3}}{9} - \frac {e^{3} x^{4} \operatorname {Li}_{1}\left (a c + b c x\right )}{16} + \frac {e^{3} x^{4} \operatorname {Li}_{2}\left (a c + b c x\right )}{4} - \frac {e^{3} x^{4}}{64} + \frac {d^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{b c} - \frac {3 d^{2} e x}{4 b c} - \frac {d e^{2} x^{2}}{6 b c} - \frac {e^{3} x^{3}}{48 b c} + \frac {3 d^{2} e \operatorname {Li}_{1}\left (a c + b c x\right )}{4 b^{2} c^{2}} - \frac {d e^{2} x}{3 b^{2} c^{2}} - \frac {e^{3} x^{2}}{32 b^{2} c^{2}} + \frac {d e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{3 b^{3} c^{3}} - \frac {e^{3} x}{16 b^{3} c^{3}} + \frac {e^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{16 b^{4} c^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*polylog(2,c*(b*x+a)),x)

[Out]

Piecewise((0, Eq(c, 0) & (Eq(b, 0) | Eq(c, 0))), ((d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4)*polyl
og(2, a*c), Eq(b, 0)), (25*a**4*e**3*polylog(1, a*c + b*c*x)/(48*b**4) - a**4*e**3*polylog(2, a*c + b*c*x)/(4*
b**4) - 11*a**3*d*e**2*polylog(1, a*c + b*c*x)/(6*b**3) + a**3*d*e**2*polylog(2, a*c + b*c*x)/b**3 + a**3*e**3
*x*polylog(1, a*c + b*c*x)/(4*b**3) + 25*a**3*e**3*x/(48*b**3) - a**3*e**3*polylog(1, a*c + b*c*x)/(b**4*c) +
9*a**2*d**2*e*polylog(1, a*c + b*c*x)/(4*b**2) - 3*a**2*d**2*e*polylog(2, a*c + b*c*x)/(2*b**2) - a**2*d*e**2*
x*polylog(1, a*c + b*c*x)/b**2 - 11*a**2*d*e**2*x/(6*b**2) - a**2*e**3*x**2*polylog(1, a*c + b*c*x)/(8*b**2) -
13*a**2*e**3*x**2/(96*b**2) + 3*a**2*d*e**2*polylog(1, a*c + b*c*x)/(b**3*c) - 23*a**2*e**3*x/(48*b**3*c) + 3
*a**2*e**3*polylog(1, a*c + b*c*x)/(4*b**4*c**2) - a*d**3*polylog(1, a*c + b*c*x)/b + a*d**3*polylog(2, a*c +
b*c*x)/b + 3*a*d**2*e*x*polylog(1, a*c + b*c*x)/(2*b) + 9*a*d**2*e*x/(4*b) + a*d*e**2*x**2*polylog(1, a*c + b*
c*x)/(2*b) + 5*a*d*e**2*x**2/(12*b) + a*e**3*x**3*polylog(1, a*c + b*c*x)/(12*b) + 7*a*e**3*x**3/(144*b) - 3*a
*d**2*e*polylog(1, a*c + b*c*x)/(b**2*c) + 7*a*d*e**2*x/(6*b**2*c) + 5*a*e**3*x**2/(48*b**2*c) - 3*a*d*e**2*po
lylog(1, a*c + b*c*x)/(2*b**3*c**2) + 13*a*e**3*x/(48*b**3*c**2) - a*e**3*polylog(1, a*c + b*c*x)/(3*b**4*c**3
) - d**3*x*polylog(1, a*c + b*c*x) + d**3*x*polylog(2, a*c + b*c*x) - d**3*x - 3*d**2*e*x**2*polylog(1, a*c +
b*c*x)/4 + 3*d**2*e*x**2*polylog(2, a*c + b*c*x)/2 - 3*d**2*e*x**2/8 - d*e**2*x**3*polylog(1, a*c + b*c*x)/3 +
d*e**2*x**3*polylog(2, a*c + b*c*x) - d*e**2*x**3/9 - e**3*x**4*polylog(1, a*c + b*c*x)/16 + e**3*x**4*polylo
g(2, a*c + b*c*x)/4 - e**3*x**4/64 + d**3*polylog(1, a*c + b*c*x)/(b*c) - 3*d**2*e*x/(4*b*c) - d*e**2*x**2/(6*
b*c) - e**3*x**3/(48*b*c) + 3*d**2*e*polylog(1, a*c + b*c*x)/(4*b**2*c**2) - d*e**2*x/(3*b**2*c**2) - e**3*x**
2/(32*b**2*c**2) + d*e**2*polylog(1, a*c + b*c*x)/(3*b**3*c**3) - e**3*x/(16*b**3*c**3) + e**3*polylog(1, a*c
+ b*c*x)/(16*b**4*c**4), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*polylog(2,c*(b*x+a)),x, algorithm="giac")

[Out]

integrate((e*x + d)^3*dilog((b*x + a)*c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, c*(a + b*x))*(d + e*x)^3,x)

[Out]

int(polylog(2, c*(a + b*x))*(d + e*x)^3, x)

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