3.2.44 \(\int \frac {\text {PolyLog}(2,c (a+b x))}{(d+e x)^4} \, dx\) [144]

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

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 448, normalized size of antiderivative = 1.00, 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 = {6733, 2465, 2440, 2438, 2442, 46, 36, 31, 2441} \begin {gather*} \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^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2\left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{3 e (b d-a e)^3}+\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 c}{6 e (d+e x) (b d-a e) (-a c e+b c d+e)}-\frac {b^2 \log (-a c-b c x+1)}{3 e (d+e x) (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \log (-a c-b c x+1)}{6 e (d+e x)^2 (b d-a e)} \end {gather*}

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 31

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

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 46

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] && Lt
Q[m + n + 2, 0])

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 2441

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]

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 \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 \text {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 \text {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*}

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Mathematica [A]
time = 0.39, size = 313, normalized size = 0.70 \begin {gather*} \frac {-\frac {2 \text {PolyLog}(2,c (a+b x))}{(d+e x)^3}+\frac {b \left (-\frac {(b d-a e)^2 \log (1-a c-b c x)}{(d+e x)^2}-\frac {2 b (b d-a e) \log (1-a c-b c x)}{d+e x}+\frac {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}+\frac {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 \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )+2 b^2 \text {PolyLog}(2,c (a+b x))+2 b^2 \text {PolyLog}\left (2,\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )\right )}{(b d-a e)^3}}{6 e} \end {gather*}

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 [A]
time = 2.14, size = 648, normalized size = 1.45

method	result	size
derivativedivides	\(\frac {\frac {c^{4} b^{4} \polylog \left (2, x b c +a c \right )}{3 \left (a e c -b c d -e \left (x b c +a c \right )\right )^{3} e}+\frac {c^{4} b^{4} \left (-\frac {\left (\frac {a c}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {d c b}{2 \left (a e c -b c d -e \right )^{2} e \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {1}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{2 \left (a e c -b c d -e \right )^{2} e}+\frac {\ln \left (-x b c -a c +1\right ) \left (2 a e c -2 b c d +e \left (-x b c -a c +1\right )-2 e \right ) \left (-x b c -a c +1\right )}{2 \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )^{2} \left (a e c -b c d -e \right )^{2}}\right ) e}{c \left (a e -b d \right )}-\frac {\left (-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-x b c -a c +1\right ) \left (-x b c -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}\right ) e}{c^{2} \left (a e -b d \right )^{2}}-\frac {\left (\frac {\dilog \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-x b c -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right ) e}{c^{3} \left (a e -b d \right )^{3}}-\frac {\dilog \left (-x b c -a c +1\right )}{c^{3} \left (a e -b d \right )^{3}}\right )}{3 e}}{b c}\)	\(648\)
default	\(\frac {\frac {c^{4} b^{4} \polylog \left (2, x b c +a c \right )}{3 \left (a e c -b c d -e \left (x b c +a c \right )\right )^{3} e}+\frac {c^{4} b^{4} \left (-\frac {\left (\frac {a c}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {d c b}{2 \left (a e c -b c d -e \right )^{2} e \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {1}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{2 \left (a e c -b c d -e \right )^{2} e}+\frac {\ln \left (-x b c -a c +1\right ) \left (2 a e c -2 b c d +e \left (-x b c -a c +1\right )-2 e \right ) \left (-x b c -a c +1\right )}{2 \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )^{2} \left (a e c -b c d -e \right )^{2}}\right ) e}{c \left (a e -b d \right )}-\frac {\left (-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-x b c -a c +1\right ) \left (-x b c -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}\right ) e}{c^{2} \left (a e -b d \right )^{2}}-\frac {\left (\frac {\dilog \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-x b c -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right ) e}{c^{3} \left (a e -b d \right )^{3}}-\frac {\dilog \left (-x b c -a c +1\right )}{c^{3} \left (a e -b d \right )^{3}}\right )}{3 e}}{b c}\)	\(648\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1519 vs. \(2 (443) = 886\).
time = 0.33, size = 1519, normalized size = 3.39 \begin {gather*} -\frac {{\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 )} b^{3}}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (\frac {b c x e + a c e - e}{b c d - a c e + e} + 1\right ) + {\rm Li}_2\left (-\frac {b c x e + a c e - e}{b c d - a c e + e}\right )\right )} b^{3}}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c^{2} e + 2 \, b^{3} c e\right )} \log \left (x e + d\right )}{6 \, {\left (b^{4} c^{2} d^{4} e + a^{4} c^{2} e^{5} - 2 \, a^{3} c e^{5} - 2 \, {\left (2 \, a b^{3} c^{2} e^{2} - b^{3} c e^{2}\right )} d^{3} + {\left (6 \, a^{2} b^{2} c^{2} e^{3} - 6 \, a b^{2} c e^{3} + b^{2} e^{3}\right )} d^{2} + a^{2} e^{5} - 2 \, {\left (2 \, a^{3} b c^{2} e^{4} - 3 \, a^{2} b c e^{4} + a b e^{4}\right )} d\right )}} + \frac {b^{4} c^{2} d^{4} - {\left (2 \, a b^{3} c^{2} e - b^{3} c e\right )} d^{3} + {\left (a^{2} b^{2} c^{2} e^{2} - a b^{2} c e^{2}\right )} d^{2} + {\left (b^{4} c^{2} d^{2} e^{2} + a^{2} b^{2} c^{2} e^{4} - a b^{2} c e^{4} - {\left (2 \, a b^{3} c^{2} e^{3} - b^{3} c e^{3}\right )} d\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d^{3} e - {\left (2 \, a b^{3} c^{2} e^{2} - b^{3} c e^{2}\right )} d^{2} + {\left (a^{2} b^{2} c^{2} e^{3} - a b^{2} c e^{3}\right )} d\right )} x - 2 \, {\left (b^{4} c^{2} d^{4} + a^{4} c^{2} e^{4} - 2 \, a^{3} c e^{4} - 2 \, {\left (2 \, a b^{3} c^{2} e - b^{3} c e\right )} d^{3} + {\left (6 \, a^{2} b^{2} c^{2} e^{2} - 6 \, a b^{2} c e^{2} + b^{2} e^{2}\right )} d^{2} + a^{2} e^{4} - 2 \, {\left (2 \, a^{3} b c^{2} e^{3} - 3 \, a^{2} b c e^{3} + a b e^{3}\right )} d\right )} {\rm Li}_2\left (b c x + a c\right ) + {\left (4 \, {\left (a b^{3} c^{2} e - b^{3} c e\right )} d^{3} + {\left (3 \, b^{4} c^{2} d e^{3} - 3 \, a b^{3} c^{2} e^{4} + 2 \, b^{3} c e^{4}\right )} x^{3} - {\left (5 \, a^{2} b^{2} c^{2} e^{2} - 8 \, a b^{2} c e^{2} + 3 \, b^{2} e^{2}\right )} d^{2} + {\left (7 \, b^{4} c^{2} d^{2} e^{2} - 2 \, a^{2} b^{2} c^{2} e^{4} + 4 \, a b^{2} c e^{4} - 2 \, b^{2} e^{4} - {\left (5 \, a b^{3} c^{2} e^{3} - 2 \, b^{3} c e^{3}\right )} d\right )} x^{2} + {\left (a^{3} b c^{2} e^{3} - 2 \, a^{2} b c e^{3} + a b e^{3}\right )} d + {\left (4 \, b^{4} c^{2} d^{3} e + a^{3} b c^{2} e^{4} - 2 \, a^{2} b c e^{4} + 2 \, {\left (a b^{3} c^{2} e^{2} - 2 \, b^{3} c e^{2}\right )} d^{2} + a b e^{4} - {\left (7 \, a^{2} b^{2} c^{2} e^{3} - 12 \, a b^{2} c e^{3} + 5 \, b^{2} e^{3}\right )} d\right )} x\right )} \log \left (-b c x - a c + 1\right )}{6 \, {\left (b^{4} c^{2} d^{7} e - 2 \, {\left (2 \, a b^{3} c^{2} e^{2} - b^{3} c e^{2}\right )} d^{6} + {\left (6 \, a^{2} b^{2} c^{2} e^{3} - 6 \, a b^{2} c e^{3} + b^{2} e^{3}\right )} d^{5} - 2 \, {\left (2 \, a^{3} b c^{2} e^{4} - 3 \, a^{2} b c e^{4} + a b e^{4}\right )} d^{4} + {\left (a^{4} c^{2} e^{5} - 2 \, a^{3} c e^{5} + a^{2} e^{5}\right )} d^{3} + {\left (b^{4} c^{2} d^{4} e^{4} + a^{4} c^{2} e^{8} - 2 \, a^{3} c e^{8} - 2 \, {\left (2 \, a b^{3} c^{2} e^{5} - b^{3} c e^{5}\right )} d^{3} + {\left (6 \, a^{2} b^{2} c^{2} e^{6} - 6 \, a b^{2} c e^{6} + b^{2} e^{6}\right )} d^{2} + a^{2} e^{8} - 2 \, {\left (2 \, a^{3} b c^{2} e^{7} - 3 \, a^{2} b c e^{7} + a b e^{7}\right )} d\right )} x^{3} + 3 \, {\left (b^{4} c^{2} d^{5} e^{3} - 2 \, {\left (2 \, a b^{3} c^{2} e^{4} - b^{3} c e^{4}\right )} d^{4} + {\left (6 \, a^{2} b^{2} c^{2} e^{5} - 6 \, a b^{2} c e^{5} + b^{2} e^{5}\right )} d^{3} - 2 \, {\left (2 \, a^{3} b c^{2} e^{6} - 3 \, a^{2} b c e^{6} + a b e^{6}\right )} d^{2} + {\left (a^{4} c^{2} e^{7} - 2 \, a^{3} c e^{7} + a^{2} e^{7}\right )} d\right )} x^{2} + 3 \, {\left (b^{4} c^{2} d^{6} e^{2} - 2 \, {\left (2 \, a b^{3} c^{2} e^{3} - b^{3} c e^{3}\right )} d^{5} + {\left (6 \, a^{2} b^{2} c^{2} e^{4} - 6 \, a b^{2} c e^{4} + b^{2} e^{4}\right )} d^{4} - 2 \, {\left (2 \, a^{3} b c^{2} e^{5} - 3 \, a^{2} b c e^{5} + a b e^{5}\right )} d^{3} + {\left (a^{4} c^{2} e^{6} - 2 \, a^{3} c e^{6} + a^{2} e^{6}\right )} d^{2}\right )} x\right )}} \end {gather*}

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

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, 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(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{{\left (d+e\,x\right )}^4} \,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)^4,x)

[Out]

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

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