3.2.38 \(\int (d+e x)^2 \text {PolyLog}(2,c (a+b x)) \, dx\) [138]

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

[Out]

-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

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Rubi [A]
time = 0.24, antiderivative size = 385, normalized size of antiderivative = 1.00, 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 = {6733, 2465, 2436, 2332, 2440, 2438, 2442, 45} \begin {gather*} -\frac {(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{9 b^3 c^3 e}-\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 {(b d-a e)^3 \text {Li}_2(c (a+b x))}{3 b^3 e}-\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 (-a c e+b c d+e)^2}{9 b^2 c^2}-\frac {x (b d-a e) (-a c e+b c d+e)}{6 b^2 c}-\frac {x (b d-a e)^2}{3 b^2}+\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((b*d - a*e)^2*x)/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 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)^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 \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b^3 c}+\frac {(b d-a e)^3 \text {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*}

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Mathematica [A]
time = 0.12, size = 274, normalized size = 0.71 \begin {gather*} \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 ) \text {PolyLog}(2,c (a+b x))}{108 b^3 c^3} \end {gather*}

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 = 0.80, size = 685, normalized size = 1.78

method	result	size
derivativedivides	\(\frac {-\frac {c \,e^{2} \polylog \left (2, x b c +a c \right ) a^{3}}{3 b^{2}}+\frac {c e \polylog \left (2, x b c +a c \right ) a^{2} d}{b}-c \polylog \left (2, x b c +a c \right ) a \,d^{2}+\frac {c b \polylog \left (2, x b c +a c \right ) d^{3}}{3 e}+\frac {e^{2} \polylog \left (2, x b c +a c \right ) a^{2} \left (x b c +a c \right )}{b^{2}}-\frac {2 e \polylog \left (2, x b c +a c \right ) a d \left (x b c +a c \right )}{b}+\polylog \left (2, x b c +a c \right ) d^{2} \left (x b c +a c \right )-\frac {e^{2} \polylog \left (2, x b c +a c \right ) a \left (x b c +a c \right )^{2}}{c \,b^{2}}+\frac {e \polylog \left (2, x b c +a c \right ) d \left (x b c +a c \right )^{2}}{c b}+\frac {e^{2} \polylog \left (2, x b c +a c \right ) \left (x b c +a c \right )^{3}}{3 c^{2} b^{2}}-\frac {\left (\frac {\left (-x b c -a c +1\right )^{3} \ln \left (-x b c -a c +1\right )}{3}-\frac {\left (-x b c -a c +1\right )^{3}}{9}\right ) e^{3}-\left (\frac {\left (-x b c -a c +1\right )^{2} \ln \left (-x b c -a c +1\right )}{2}-\frac {\left (-x b c -a c +1\right )^{2}}{4}\right ) \left (-3 a c \,e^{3}+3 b c d \,e^{2}+2 e^{3}\right )+3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) a^{2} c^{2} e^{3}-6 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) a b \,c^{2} d \,e^{2}+3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) b^{2} c^{2} d^{2} e -3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) a c \,e^{3}+3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) b c d \,e^{2}+\left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) e^{3}-\dilog \left (-x b c -a c +1\right ) c^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 c^{2} b^{2} e}}{b c}\)	\(685\)
default	\(\frac {-\frac {c \,e^{2} \polylog \left (2, x b c +a c \right ) a^{3}}{3 b^{2}}+\frac {c e \polylog \left (2, x b c +a c \right ) a^{2} d}{b}-c \polylog \left (2, x b c +a c \right ) a \,d^{2}+\frac {c b \polylog \left (2, x b c +a c \right ) d^{3}}{3 e}+\frac {e^{2} \polylog \left (2, x b c +a c \right ) a^{2} \left (x b c +a c \right )}{b^{2}}-\frac {2 e \polylog \left (2, x b c +a c \right ) a d \left (x b c +a c \right )}{b}+\polylog \left (2, x b c +a c \right ) d^{2} \left (x b c +a c \right )-\frac {e^{2} \polylog \left (2, x b c +a c \right ) a \left (x b c +a c \right )^{2}}{c \,b^{2}}+\frac {e \polylog \left (2, x b c +a c \right ) d \left (x b c +a c \right )^{2}}{c b}+\frac {e^{2} \polylog \left (2, x b c +a c \right ) \left (x b c +a c \right )^{3}}{3 c^{2} b^{2}}-\frac {\left (\frac {\left (-x b c -a c +1\right )^{3} \ln \left (-x b c -a c +1\right )}{3}-\frac {\left (-x b c -a c +1\right )^{3}}{9}\right ) e^{3}-\left (\frac {\left (-x b c -a c +1\right )^{2} \ln \left (-x b c -a c +1\right )}{2}-\frac {\left (-x b c -a c +1\right )^{2}}{4}\right ) \left (-3 a c \,e^{3}+3 b c d \,e^{2}+2 e^{3}\right )+3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) a^{2} c^{2} e^{3}-6 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) a b \,c^{2} d \,e^{2}+3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) b^{2} c^{2} d^{2} e -3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) a c \,e^{3}+3 \left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) b c d \,e^{2}+\left (\left (-x b c -a c +1\right ) \ln \left (-x b c -a c +1\right )-1+x b c +a c \right ) e^{3}-\dilog \left (-x b c -a c +1\right ) c^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 c^{2} b^{2} e}}{b c}\)	\(685\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/c*(-1/3*c/b^2*e^2*polylog(2,b*c*x+a*c)*a^3+c/b*e*polylog(2,b*c*x+a*c)*a^2*d-c*polylog(2,b*c*x+a*c)*a*d^2+1
/3*c*b/e*polylog(2,b*c*x+a*c)*d^3+1/b^2*e^2*polylog(2,b*c*x+a*c)*a^2*(b*c*x+a*c)-2/b*e*polylog(2,b*c*x+a*c)*a*
d*(b*c*x+a*c)+polylog(2,b*c*x+a*c)*d^2*(b*c*x+a*c)-1/c/b^2*e^2*polylog(2,b*c*x+a*c)*a*(b*c*x+a*c)^2+1/c/b*e*po
lylog(2,b*c*x+a*c)*d*(b*c*x+a*c)^2+1/3/c^2/b^2*e^2*polylog(2,b*c*x+a*c)*(b*c*x+a*c)^3-1/3/c^2/b^2/e*((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)*e^3-(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^3+3*b*c*d*e^2+2*e^3)+3*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a^2*c^2*e^3-6*((-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^2+3*((-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-3*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*a*c*e^3+3*((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*b*c*
d*e^2+((-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)-1+x*b*c+a*c)*e^3-dilog(-b*c*x-a*c+1)*c^3*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2
*d^2*e-b^3*d^3)))

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Maxima [A]
time = 0.27, size = 416, normalized size = 1.08 \begin {gather*} -\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} x^{3} e^{2} + 3 \, {\left (9 \, b^{3} c^{3} d e - 5 \, a b^{2} c^{3} e^{2} + 2 \, b^{2} c^{2} e^{2}\right )} x^{2} + 6 \, {\left (18 \, b^{3} c^{3} d^{2} + 11 \, a^{2} b c^{3} e^{2} - 7 \, a b c^{2} e^{2} + 2 \, b c e^{2} - 9 \, {\left (3 \, a b^{2} c^{3} e - b^{2} c^{2} e\right )} d\right )} x - 36 \, {\left (b^{3} c^{3} x^{3} e^{2} + 3 \, b^{3} c^{3} d x^{2} e + 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} x^{3} e^{2} + 11 \, a^{3} c^{3} e^{2} - 18 \, a^{2} c^{2} e^{2} + 18 \, {\left (a b^{2} c^{3} - b^{2} c^{2}\right )} d^{2} + 3 \, {\left (3 \, b^{3} c^{3} d e - a b^{2} c^{3} e^{2}\right )} x^{2} + 9 \, a c e^{2} - 9 \, {\left (3 \, a^{2} b c^{3} e - 4 \, a b c^{2} e + b c e\right )} d + 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 - 2 \, e^{2}\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \end {gather*}

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*x^3*e^2 + 3*(9*b^3*c^3*d*e - 5*a*b^2*c^3*e^2 + 2*b^2*c^2*e^2)*x^2 + 6*(18*b^3*c^3*d^2 +
11*a^2*b*c^3*e^2 - 7*a*b*c^2*e^2 + 2*b*c*e^2 - 9*(3*a*b^2*c^3*e - b^2*c^2*e)*d)*x - 36*(b^3*c^3*x^3*e^2 + 3*b
^3*c^3*d*x^2*e + 3*b^3*c^3*d^2*x)*dilog(b*c*x + a*c) - 6*(2*b^3*c^3*x^3*e^2 + 11*a^3*c^3*e^2 - 18*a^2*c^2*e^2
+ 18*(a*b^2*c^3 - b^2*c^2)*d^2 + 3*(3*b^3*c^3*d*e - a*b^2*c^3*e^2)*x^2 + 9*a*c*e^2 - 9*(3*a^2*b*c^3*e - 4*a*b*
c^2*e + b*c*e)*d + 6*(3*b^3*c^3*d^2 - 3*a*b^2*c^3*d*e + a^2*b*c^3*e^2)*x - 2*e^2)*log(-b*c*x - a*c + 1))/(b^3*
c^3)

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Fricas [A]
time = 0.35, size = 361, normalized size = 0.94 \begin {gather*} -\frac {108 \, b^{3} c^{3} d^{2} x - 36 \, {\left (3 \, b^{3} c^{3} d^{2} x + 3 \, a b^{2} c^{3} d^{2} + {\left (b^{3} c^{3} x^{3} + a^{3} c^{3}\right )} e^{2} + 3 \, {\left (b^{3} c^{3} d x^{2} - a^{2} b c^{3} d\right )} e\right )} {\rm Li}_2\left (b c x + a c\right ) + {\left (4 \, b^{3} c^{3} x^{3} - 3 \, {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} x^{2} + 6 \, {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} x\right )} e^{2} + 27 \, {\left (b^{3} c^{3} d x^{2} - 2 \, {\left (3 \, a b^{2} c^{3} - b^{2} c^{2}\right )} d x\right )} e - 6 \, {\left (18 \, b^{3} c^{3} d^{2} x + 18 \, {\left (a b^{2} c^{3} - b^{2} c^{2}\right )} d^{2} + {\left (2 \, b^{3} c^{3} x^{3} - 3 \, a b^{2} c^{3} x^{2} + 6 \, a^{2} b c^{3} x + 11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} e^{2} + 9 \, {\left (b^{3} c^{3} d x^{2} - 2 \, a b^{2} c^{3} d x - {\left (3 \, a^{2} b c^{3} - 4 \, a b c^{2} + b c\right )} d\right )} e\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \end {gather*}

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*(108*b^3*c^3*d^2*x - 36*(3*b^3*c^3*d^2*x + 3*a*b^2*c^3*d^2 + (b^3*c^3*x^3 + a^3*c^3)*e^2 + 3*(b^3*c^3*d
*x^2 - a^2*b*c^3*d)*e)*dilog(b*c*x + a*c) + (4*b^3*c^3*x^3 - 3*(5*a*b^2*c^3 - 2*b^2*c^2)*x^2 + 6*(11*a^2*b*c^3
- 7*a*b*c^2 + 2*b*c)*x)*e^2 + 27*(b^3*c^3*d*x^2 - 2*(3*a*b^2*c^3 - b^2*c^2)*d*x)*e - 6*(18*b^3*c^3*d^2*x + 18
*(a*b^2*c^3 - b^2*c^2)*d^2 + (2*b^3*c^3*x^3 - 3*a*b^2*c^3*x^2 + 6*a^2*b*c^3*x + 11*a^3*c^3 - 18*a^2*c^2 + 9*a*
c - 2)*e^2 + 9*(b^3*c^3*d*x^2 - 2*a*b^2*c^3*d*x - (3*a^2*b*c^3 - 4*a*b*c^2 + b*c)*d)*e)*log(-b*c*x - a*c + 1))
/(b^3*c^3)

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Sympy [A]
time = 10.21, size = 561, normalized size = 1.46 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((0, Eq(b, 0) & Eq(c, 0)), ((d**2*x + d*e*x**2 + e**2*x**3/3)*polylog(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
*polylog(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))

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

[Out]

integrate((e*x + d)^2*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 )}^2 \,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)^2,x)

[Out]

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

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