3.144 \(\int \frac{\text{PolyLog}(2,c (a+b x))}{(d+e x)^4} \, dx\)

Optimal. Leaf size=448 \[ \frac{b^3 \text{PolyLog}(2,c (a+b x))}{3 e (b d-a e)^3}+\frac{b^3 \text{PolyLog}\left (2,\frac{e (-a c-b c x+1)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}-\frac{\text{PolyLog}(2,c (a+b x))}{3 e (d+e x)^3}+\frac{b^3 c^2 \log (-a c-b c x+1)}{6 e (b d-a e) (-a c e+b c d+e)^2}-\frac{b^3 c^2 \log (d+e x)}{6 e (b d-a e) (-a c e+b c d+e)^2}+\frac{b^2 c}{6 e (d+e x) (b d-a e) (-a c e+b c d+e)}+\frac{b^3 c \log (-a c-b c x+1)}{3 e (b d-a e)^2 (-a c e+b c d+e)}-\frac{b^3 c \log (d+e x)}{3 e (b d-a e)^2 (-a c e+b c d+e)}+\frac{b^3 \log (-a c-b c x+1) \log \left (\frac{b c (d+e x)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}-\frac{b^2 \log (-a c-b c x+1)}{3 e (d+e x) (b d-a e)^2}-\frac{b \log (-a c-b c x+1)}{6 e (d+e x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.426017, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {6598, 2418, 2393, 2391, 2395, 44, 36, 31, 2394} \[ \frac{b^3 \text{PolyLog}(2,c (a+b x))}{3 e (b d-a e)^3}+\frac{b^3 \text{PolyLog}\left (2,\frac{e (-a c-b c x+1)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}-\frac{\text{PolyLog}(2,c (a+b x))}{3 e (d+e x)^3}+\frac{b^3 c^2 \log (-a c-b c x+1)}{6 e (b d-a e) (-a c e+b c d+e)^2}-\frac{b^3 c^2 \log (d+e x)}{6 e (b d-a e) (-a c e+b c d+e)^2}+\frac{b^2 c}{6 e (d+e x) (b d-a e) (-a c e+b c d+e)}+\frac{b^3 c \log (-a c-b c x+1)}{3 e (b d-a e)^2 (-a c e+b c d+e)}-\frac{b^3 c \log (d+e x)}{3 e (b d-a e)^2 (-a c e+b c d+e)}+\frac{b^3 \log (-a c-b c x+1) \log \left (\frac{b c (d+e x)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}-\frac{b^2 \log (-a c-b c x+1)}{3 e (d+e x) (b d-a e)^2}-\frac{b \log (-a c-b c x+1)}{6 e (d+e x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, c*(a + b*x)]/(d + e*x)^4,x]

[Out]

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

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 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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 31

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

Rule 2394

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

Rubi steps

\begin{align*} \int \frac{\text{Li}_2(c (a+b x))}{(d+e x)^4} \, dx &=-\frac{\text{Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac{b \int \frac{\log (1-a c-b c x)}{(a+b x) (d+e x)^3} \, dx}{3 e}\\ &=-\frac{\text{Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac{b \int \left (\frac{b^3 \log (1-a c-b c x)}{(b d-a e)^3 (a+b x)}-\frac{e \log (1-a c-b c x)}{(b d-a e) (d+e x)^3}-\frac{b e \log (1-a c-b c x)}{(b d-a e)^2 (d+e x)^2}-\frac{b^2 e \log (1-a c-b c x)}{(b d-a e)^3 (d+e x)}\right ) \, dx}{3 e}\\ &=-\frac{\text{Li}_2(c (a+b x))}{3 e (d+e x)^3}+\frac{b^3 \int \frac{\log (1-a c-b c x)}{d+e x} \, dx}{3 (b d-a e)^3}-\frac{b^4 \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{3 e (b d-a e)^3}+\frac{b^2 \int \frac{\log (1-a c-b c x)}{(d+e x)^2} \, dx}{3 (b d-a e)^2}+\frac{b \int \frac{\log (1-a c-b c x)}{(d+e x)^3} \, dx}{3 (b d-a e)}\\ &=-\frac{b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac{b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}+\frac{b^3 \log (1-a c-b c x) \log \left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}-\frac{\text{Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 e (b d-a e)^3}+\frac{\left (b^4 c\right ) \int \frac{\log \left (-\frac{b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{3 e (b d-a e)^3}-\frac{\left (b^3 c\right ) \int \frac{1}{(1-a c-b c x) (d+e x)} \, dx}{3 e (b d-a e)^2}-\frac{\left (b^2 c\right ) \int \frac{1}{(1-a c-b c x) (d+e x)^2} \, dx}{6 e (b d-a e)}\\ &=-\frac{b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac{b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}+\frac{b^3 \log (1-a c-b c x) \log \left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac{b^3 \text{Li}_2(c (a+b x))}{3 e (b d-a e)^3}-\frac{\text{Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{3 e (b d-a e)^3}-\frac{\left (b^2 c\right ) \int \left (-\frac{b^2 c^2}{(b c d+e-a c e)^2 (-1+a c+b c x)}+\frac{e}{(b c d+(1-a c) e) (d+e x)^2}+\frac{b c e}{(b c d+(1-a c) e)^2 (d+e x)}\right ) \, dx}{6 e (b d-a e)}-\frac{\left (b^3 c\right ) \int \frac{1}{d+e x} \, dx}{3 (b d-a e)^2 (b c d+e-a c e)}-\frac{\left (b^4 c^2\right ) \int \frac{1}{1-a c-b c x} \, dx}{3 e (b d-a e)^2 (b c d+e-a c e)}\\ &=\frac{b^2 c}{6 e (b d-a e) (b c d+e-a c e) (d+e x)}+\frac{b^3 c^2 \log (1-a c-b c x)}{6 e (b d-a e) (b c d+e-a c e)^2}+\frac{b^3 c \log (1-a c-b c x)}{3 e (b d-a e)^2 (b c d+e-a c e)}-\frac{b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac{b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}-\frac{b^3 c^2 \log (d+e x)}{6 e (b d-a e) (b c d+e-a c e)^2}-\frac{b^3 c \log (d+e x)}{3 e (b d-a e)^2 (b c d+e-a c e)}+\frac{b^3 \log (1-a c-b c x) \log \left (\frac{b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac{b^3 \text{Li}_2(c (a+b x))}{3 e (b d-a e)^3}-\frac{\text{Li}_2(c (a+b x))}{3 e (d+e x)^3}+\frac{b^3 \text{Li}_2\left (\frac{e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}\\ \end{align*}

Mathematica [A]  time = 0.631829, size = 313, normalized size = 0.7 \[ \frac{\frac{b \left (2 b^2 \text{PolyLog}\left (2,\frac{e (a c+b c x-1)}{e (a c-1)-b c d}\right )+2 b^2 \text{PolyLog}(2,c (a+b x))+\frac{2 b^2 c (b d-a e) (\log (-a c-b c x+1)-\log (d+e x))}{-a c e+b c d+e}+2 b^2 \log (-a c-b c x+1) \log \left (\frac{b c (d+e x)}{-a c e+b c d+e}\right )-\frac{2 b (b d-a e) \log (-a c-b c x+1)}{d+e x}+\frac{b c (b d-a e)^2 (b c (d+e x) \log (-a c-b c x+1)-a c e-b c (d+e x) \log (d+e x)+b c d+e)}{(d+e x) (-a c e+b c d+e)^2}-\frac{(b d-a e)^2 \log (-a c-b c x+1)}{(d+e x)^2}\right )}{(b d-a e)^3}-\frac{2 \text{PolyLog}(2,c (a+b x))}{(d+e x)^3}}{6 e} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, c*(a + b*x)]/(d + e*x)^4,x]

[Out]

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

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Maple [B]  time = 0.394, size = 1075, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,c*(b*x+a))/(e*x+d)^4,x)

[Out]

-1/3*b^3*c^3/(b*c*e*x+b*c*d)^3/e*polylog(2,b*c*x+a*c)-1/3*b^3/e/(a*e-b*d)^3*dilog((a*c*e-b*c*d+(-b*c*x-a*c+1)*
e-e)/(a*c*e-b*c*d-e))-1/3*b^3/e/(a*e-b*d)^3*ln(-b*c*x-a*c+1)*ln((a*c*e-b*c*d+(-b*c*x-a*c+1)*e-e)/(a*c*e-b*c*d-
e))+1/3*b^3*c/e/(a*e-b*d)^2/(a*c*e-b*c*d-e)*ln(a*c*e-b*c*d+(-b*c*x-a*c+1)*e-e)+1/3*b^4*c^2/(a*e-b*d)^2*ln(-b*c
*x-a*c+1)/(a*c*e-b*c*d-e)/(-b*c*e*x-b*c*d)*x+1/3*b^3*c^2/(a*e-b*d)^2*ln(-b*c*x-a*c+1)/(a*c*e-b*c*d-e)/(-b*c*e*
x-b*c*d)*a-1/3*b^3*c/(a*e-b*d)^2*ln(-b*c*x-a*c+1)/(a*c*e-b*c*d-e)/(-b*c*e*x-b*c*d)-1/3*b^3/e/(a*e-b*d)^3*dilog
(-b*c*x-a*c+1)+1/6*b^3*c^2/e/(a*e-b*d)/(a*c*e-b*c*d-e)^2*ln(a*c*e-b*c*d+(-b*c*x-a*c+1)*e-e)-1/6*b^3*c^3/(a*e-b
*d)/(a*c*e-b*c*d-e)^2/(-b*c*e*x-b*c*d)*a+1/6*b^4*c^3/e/(a*e-b*d)/(a*c*e-b*c*d-e)^2/(-b*c*e*x-b*c*d)*d+1/6*b^3*
c^2/(a*e-b*d)/(a*c*e-b*c*d-e)^2/(-b*c*e*x-b*c*d)-1/6*b^5*c^4*e/(a*e-b*d)*ln(-b*c*x-a*c+1)/(-b*c*e*x-b*c*d)^2/(
a*c*e-b*c*d-e)^2*x^2-1/3*b^5*c^4/(a*e-b*d)*ln(-b*c*x-a*c+1)/(-b*c*e*x-b*c*d)^2/(a*c*e-b*c*d-e)^2*x*d+1/6*b^3*c
^4*e/(a*e-b*d)*ln(-b*c*x-a*c+1)/(-b*c*e*x-b*c*d)^2/(a*c*e-b*c*d-e)^2*a^2-1/3*b^4*c^4/(a*e-b*d)*ln(-b*c*x-a*c+1
)/(-b*c*e*x-b*c*d)^2/(a*c*e-b*c*d-e)^2*d*a-1/3*b^3*c^3*e/(a*e-b*d)*ln(-b*c*x-a*c+1)/(-b*c*e*x-b*c*d)^2/(a*c*e-
b*c*d-e)^2*a+1/3*b^4*c^3/(a*e-b*d)*ln(-b*c*x-a*c+1)/(-b*c*e*x-b*c*d)^2/(a*c*e-b*c*d-e)^2*d+1/6*b^3*c^2*e/(a*e-
b*d)*ln(-b*c*x-a*c+1)/(-b*c*e*x-b*c*d)^2/(a*c*e-b*c*d-e)^2

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Maxima [B]  time = 1.22441, size = 1928, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*(log(b*c*x + a*c)*log(-b*c*x - a*c + 1) + dilog(-b*c*x - a*c + 1))*b^3/(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a
^2*b*d*e^3 - a^3*e^4) + 1/3*(log(-b*c*x - a*c + 1)*log((b*c*e*x + (a*c - 1)*e)/(b*c*d - (a*c - 1)*e) + 1) + di
log(-(b*c*e*x + (a*c - 1)*e)/(b*c*d - (a*c - 1)*e)))*b^3/(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^
4) - 1/6*(3*b^4*c^2*d - (3*a*b^3*c^2 - 2*b^3*c)*e)*log(e*x + d)/(b^4*c^2*d^4*e - 2*(2*a*b^3*c^2 - b^3*c)*d^3*e
^2 + (6*a^2*b^2*c^2 - 6*a*b^2*c + b^2)*d^2*e^3 - 2*(2*a^3*b*c^2 - 3*a^2*b*c + a*b)*d*e^4 + (a^4*c^2 - 2*a^3*c
+ a^2)*e^5) + 1/6*(b^4*c^2*d^4 - (2*a*b^3*c^2 - b^3*c)*d^3*e + (a^2*b^2*c^2 - a*b^2*c)*d^2*e^2 + (b^4*c^2*d^2*
e^2 - (2*a*b^3*c^2 - b^3*c)*d*e^3 + (a^2*b^2*c^2 - a*b^2*c)*e^4)*x^2 + 2*(b^4*c^2*d^3*e - (2*a*b^3*c^2 - b^3*c
)*d^2*e^2 + (a^2*b^2*c^2 - a*b^2*c)*d*e^3)*x - 2*(b^4*c^2*d^4 - 2*(2*a*b^3*c^2 - b^3*c)*d^3*e + (6*a^2*b^2*c^2
- 6*a*b^2*c + b^2)*d^2*e^2 - 2*(2*a^3*b*c^2 - 3*a^2*b*c + a*b)*d*e^3 + (a^4*c^2 - 2*a^3*c + a^2)*e^4)*dilog(b
*c*x + a*c) + (4*(a*b^3*c^2 - b^3*c)*d^3*e - (5*a^2*b^2*c^2 - 8*a*b^2*c + 3*b^2)*d^2*e^2 + (a^3*b*c^2 - 2*a^2*
b*c + a*b)*d*e^3 + (3*b^4*c^2*d*e^3 - (3*a*b^3*c^2 - 2*b^3*c)*e^4)*x^3 + (7*b^4*c^2*d^2*e^2 - (5*a*b^3*c^2 - 2
*b^3*c)*d*e^3 - 2*(a^2*b^2*c^2 - 2*a*b^2*c + b^2)*e^4)*x^2 + (4*b^4*c^2*d^3*e + 2*(a*b^3*c^2 - 2*b^3*c)*d^2*e^
2 - (7*a^2*b^2*c^2 - 12*a*b^2*c + 5*b^2)*d*e^3 + (a^3*b*c^2 - 2*a^2*b*c + a*b)*e^4)*x)*log(-b*c*x - a*c + 1))/
(b^4*c^2*d^7*e - 2*(2*a*b^3*c^2 - b^3*c)*d^6*e^2 + (6*a^2*b^2*c^2 - 6*a*b^2*c + b^2)*d^5*e^3 - 2*(2*a^3*b*c^2
- 3*a^2*b*c + a*b)*d^4*e^4 + (a^4*c^2 - 2*a^3*c + a^2)*d^3*e^5 + (b^4*c^2*d^4*e^4 - 2*(2*a*b^3*c^2 - b^3*c)*d^
3*e^5 + (6*a^2*b^2*c^2 - 6*a*b^2*c + b^2)*d^2*e^6 - 2*(2*a^3*b*c^2 - 3*a^2*b*c + a*b)*d*e^7 + (a^4*c^2 - 2*a^3
*c + a^2)*e^8)*x^3 + 3*(b^4*c^2*d^5*e^3 - 2*(2*a*b^3*c^2 - b^3*c)*d^4*e^4 + (6*a^2*b^2*c^2 - 6*a*b^2*c + b^2)*
d^3*e^5 - 2*(2*a^3*b*c^2 - 3*a^2*b*c + a*b)*d^2*e^6 + (a^4*c^2 - 2*a^3*c + a^2)*d*e^7)*x^2 + 3*(b^4*c^2*d^6*e^
2 - 2*(2*a*b^3*c^2 - b^3*c)*d^5*e^3 + (6*a^2*b^2*c^2 - 6*a*b^2*c + b^2)*d^4*e^4 - 2*(2*a^3*b*c^2 - 3*a^2*b*c +
a*b)*d^3*e^5 + (a^4*c^2 - 2*a^3*c + a^2)*d^2*e^6)*x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm Li}_2\left (b c x + a c\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral(dilog(b*c*x + a*c)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

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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(polylog(2,c*(b*x+a))/(e*x+d)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/(e*x+d)^4,x, algorithm="giac")

[Out]

integrate(dilog((b*x + a)*c)/(e*x + d)^4, x)