3.233 \(\int \frac{(a+b \log (c (d+e x)^n))^3}{(f+g x) (h+i x)^2} \, dx\)

Optimal. Leaf size=602 \[ \frac{6 b^2 e n^2 \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e h-d i) (g h-f i)}-\frac{6 b^2 g n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac{6 b^2 g n^2 \text{PolyLog}\left (3,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac{3 b g n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac{3 b g n \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac{6 b^3 e n^3 \text{PolyLog}\left (3,-\frac{i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}+\frac{6 b^3 g n^3 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac{6 b^3 g n^3 \text{PolyLog}\left (4,-\frac{i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}+\frac{3 b e n \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e h-d i) (g h-f i)}-\frac{i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+i x) (e h-d i) (g h-f i)}+\frac{g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2}-\frac{g \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2} \]

[Out]

-((i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*h - d*i)*(g*h - f*i)*(h + i*x))) + (g*(a + b*Log[c*(d + e*x)^
n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^2 + (3*b*e*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(h + i*x))/
(e*h - d*i)])/((e*h - d*i)*(g*h - f*i)) - (g*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h
- f*i)^2 + (3*b*g*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 + (6
*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) - (3
*b*g*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 - (6*b^2*g*n^2*(a
+ b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 - (6*b^3*e*n^3*PolyLog[3, -((i
*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) + (6*b^2*g*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((i
*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 + (6*b^3*g*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^
2 - (6*b^3*g*n^3*PolyLog[4, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2

________________________________________________________________________________________

Rubi [A]  time = 0.725129, antiderivative size = 602, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2418, 2396, 2433, 2374, 2383, 6589, 2397} \[ \frac{6 b^2 e n^2 \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e h-d i) (g h-f i)}-\frac{6 b^2 g n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac{6 b^2 g n^2 \text{PolyLog}\left (3,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac{3 b g n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac{3 b g n \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac{6 b^3 e n^3 \text{PolyLog}\left (3,-\frac{i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}+\frac{6 b^3 g n^3 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac{6 b^3 g n^3 \text{PolyLog}\left (4,-\frac{i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}+\frac{3 b e n \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e h-d i) (g h-f i)}-\frac{i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+i x) (e h-d i) (g h-f i)}+\frac{g \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2}-\frac{g \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^3/((f + g*x)*(h + i*x)^2),x]

[Out]

-((i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*h - d*i)*(g*h - f*i)*(h + i*x))) + (g*(a + b*Log[c*(d + e*x)^
n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^2 + (3*b*e*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(h + i*x))/
(e*h - d*i)])/((e*h - d*i)*(g*h - f*i)) - (g*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h
- f*i)^2 + (3*b*g*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 + (6
*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) - (3
*b*g*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 - (6*b^2*g*n^2*(a
+ b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 - (6*b^3*e*n^3*PolyLog[3, -((i
*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) + (6*b^2*g*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((i
*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 + (6*b^3*g*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^
2 - (6*b^3*g*n^3*PolyLog[4, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2

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 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+233 x)^2 (f+g x)} \, dx &=\int \left (\frac{233 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h) (h+233 x)^2}-\frac{233 g \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2 (h+233 x)}+\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2 (f+g x)}\right ) \, dx\\ &=-\frac{(233 g) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{h+233 x} \, dx}{(233 f-g h)^2}+\frac{g^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{(233 f-g h)^2}+\frac{233 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+233 x)^2} \, dx}{233 f-g h}\\ &=-\frac{233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac{g \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{(3 b e g n) \int \frac{\log \left (\frac{e (h+233 x)}{-233 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx}{(233 f-g h)^2}-\frac{(3 b e g n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{(233 f-g h)^2}+\frac{(699 b e n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{h+233 x} \, dx}{(233 d-e h) (233 f-g h)}\\ &=\frac{3 b e n \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac{233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac{g \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{(3 b g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{e \left (\frac{-233 d+e h}{e}+\frac{233 x}{e}\right )}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac{(3 b g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{e \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac{\left (6 b^2 e^2 n^2\right ) \int \frac{\log \left (\frac{e (h+233 x)}{-233 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{(233 d-e h) (233 f-g h)}\\ &=\frac{3 b e n \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac{233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac{g \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}-\frac{3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}-\frac{\left (6 b^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}+\frac{\left (6 b^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{233 x}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac{\left (6 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{-233 d+e h}{e}+\frac{233 x}{e}\right )}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 d-e h) (233 f-g h)}\\ &=\frac{3 b e n \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac{233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac{g \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 d-e h) (233 f-g h)}-\frac{3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}-\frac{6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}+\frac{\left (6 b^3 g n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac{\left (6 b^3 g n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{233 x}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac{\left (6 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{233 x}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 d-e h) (233 f-g h)}\\ &=\frac{3 b e n \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac{233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac{g \log \left (-\frac{e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}+\frac{6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 d-e h) (233 f-g h)}-\frac{3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}-\frac{6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}-\frac{6 b^3 e n^3 \text{Li}_3\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 d-e h) (233 f-g h)}+\frac{6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}+\frac{6 b^3 g n^3 \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}-\frac{6 b^3 g n^3 \text{Li}_4\left (\frac{233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}\\ \end{align*}

Mathematica [A]  time = 1.43843, size = 1025, normalized size = 1.7 \[ \frac{-b^3 \left ((g h-f i) \left (\left (i (d+e x) \log (d+e x)-3 e (h+i x) \log \left (\frac{e (h+i x)}{e h-d i}\right )\right ) \log ^2(d+e x)-6 e (h+i x) \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right ) \log (d+e x)+6 e (h+i x) \text{PolyLog}\left (3,\frac{i (d+e x)}{d i-e h}\right )\right )-g (e h-d i) (h+i x) \left (\log \left (\frac{e (f+g x)}{e f-d g}\right ) \log ^3(d+e x)+3 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) \log ^2(d+e x)-6 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right ) \log (d+e x)+6 \text{PolyLog}\left (4,\frac{g (d+e x)}{d g-e f}\right )\right )+g (e h-d i) (h+i x) \left (\log \left (\frac{e (h+i x)}{e h-d i}\right ) \log ^3(d+e x)+3 \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right ) \log ^2(d+e x)-6 \text{PolyLog}\left (3,\frac{i (d+e x)}{d i-e h}\right ) \log (d+e x)+6 \text{PolyLog}\left (4,\frac{i (d+e x)}{d i-e h}\right )\right )\right ) n^3-3 b^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left ((g h-f i) \left (\log (d+e x) \left (i (d+e x) \log (d+e x)-2 e (h+i x) \log \left (\frac{e (h+i x)}{e h-d i}\right )\right )-2 e (h+i x) \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right )\right )-g (e h-d i) (h+i x) \left (\log \left (\frac{e (f+g x)}{e f-d g}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )\right )+g (e h-d i) (h+i x) \left (\log \left (\frac{e (h+i x)}{e h-d i}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{i (d+e x)}{d i-e h}\right )\right )\right ) n^2-3 b \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left ((g h-f i) (i (d+e x) \log (d+e x)-e (h+i x) \log (h+i x))-g (e h-d i) (h+i x) \left (\log (d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )+\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )\right )+g (e h-d i) (h+i x) \left (\log (d+e x) \log \left (\frac{e (h+i x)}{e h-d i}\right )+\text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right )\right )\right ) n+(e h-d i) (g h-f i) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+g (e h-d i) (h+i x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)-g (e h-d i) (h+i x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (h+i x)}{(e h-d i) (g h-f i)^2 (h+i x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^3/((f + g*x)*(h + i*x)^2),x]

[Out]

((e*h - d*i)*(g*h - f*i)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3 + g*(e*h - d*i)*(h + i*x)*(a - b*n*Lo
g[d + e*x] + b*Log[c*(d + e*x)^n])^3*Log[f + g*x] - g*(e*h - d*i)*(h + i*x)*(a - b*n*Log[d + e*x] + b*Log[c*(d
+ e*x)^n])^3*Log[h + i*x] - 3*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*((g*h - f*i)*(i*(d + e*x)*L
og[d + e*x] - e*(h + i*x)*Log[h + i*x]) - g*(e*h - d*i)*(h + i*x)*(Log[d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)]
+ PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + g*(e*h - d*i)*(h + i*x)*(Log[d + e*x]*Log[(e*(h + i*x))/(e*h -
d*i)] + PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)])) - 3*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(
(g*h - f*i)*(Log[d + e*x]*(i*(d + e*x)*Log[d + e*x] - 2*e*(h + i*x)*Log[(e*(h + i*x))/(e*h - d*i)]) - 2*e*(h +
i*x)*PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)]) - g*(e*h - d*i)*(h + i*x)*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e
*f - d*g)] + 2*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*
g)]) + g*(e*h - d*i)*(h + i*x)*(Log[d + e*x]^2*Log[(e*(h + i*x))/(e*h - d*i)] + 2*Log[d + e*x]*PolyLog[2, (i*(
d + e*x))/(-(e*h) + d*i)] - 2*PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)])) - b^3*n^3*((g*h - f*i)*(Log[d + e*x]^
2*(i*(d + e*x)*Log[d + e*x] - 3*e*(h + i*x)*Log[(e*(h + i*x))/(e*h - d*i)]) - 6*e*(h + i*x)*Log[d + e*x]*PolyL
og[2, (i*(d + e*x))/(-(e*h) + d*i)] + 6*e*(h + i*x)*PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)]) - g*(e*h - d*i)*
(h + i*x)*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f) +
d*g)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 6*PolyLog[4, (g*(d + e*x))/(-(e*f) + d*g)])
+ g*(e*h - d*i)*(h + i*x)*(Log[d + e*x]^3*Log[(e*(h + i*x))/(e*h - d*i)] + 3*Log[d + e*x]^2*PolyLog[2, (i*(d
+ e*x))/(-(e*h) + d*i)] - 6*Log[d + e*x]*PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)] + 6*PolyLog[4, (i*(d + e*x))
/(-(e*h) + d*i)])))/((e*h - d*i)*(g*h - f*i)^2*(h + i*x))

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Maple [F]  time = 1.974, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}{ \left ( gx+f \right ) \left ( ix+h \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h)^2,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3}{\left (\frac{g \log \left (g x + f\right )}{g^{2} h^{2} - 2 \, f g h i + f^{2} i^{2}} - \frac{g \log \left (i x + h\right )}{g^{2} h^{2} - 2 \, f g h i + f^{2} i^{2}} + \frac{1}{g h^{2} - f h i +{\left (g h i - f i^{2}\right )} x}\right )} + \int \frac{b^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + b^{3} \log \left (c\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + 3 \,{\left (b^{3} \log \left (c\right ) + a b^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 3 \,{\left (b^{3} \log \left (c\right )^{2} + 2 \, a b^{2} \log \left (c\right ) + a^{2} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g i^{2} x^{3} + f h^{2} +{\left (2 \, g h i + f i^{2}\right )} x^{2} +{\left (g h^{2} + 2 \, f h i\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h)^2,x, algorithm="maxima")

[Out]

a^3*(g*log(g*x + f)/(g^2*h^2 - 2*f*g*h*i + f^2*i^2) - g*log(i*x + h)/(g^2*h^2 - 2*f*g*h*i + f^2*i^2) + 1/(g*h^
2 - f*h*i + (g*h*i - f*i^2)*x)) + integrate((b^3*log((e*x + d)^n)^3 + b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*
b*log(c) + 3*(b^3*log(c) + a*b^2)*log((e*x + d)^n)^2 + 3*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log((e*x + d)
^n))/(g*i^2*x^3 + f*h^2 + (2*g*h*i + f*i^2)*x^2 + (g*h^2 + 2*f*h*i)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{3}}{g i^{2} x^{3} + f h^{2} +{\left (2 \, g h i + f i^{2}\right )} x^{2} +{\left (g h^{2} + 2 \, f h i\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h)^2,x, algorithm="fricas")

[Out]

integral((b^3*log((e*x + d)^n*c)^3 + 3*a*b^2*log((e*x + d)^n*c)^2 + 3*a^2*b*log((e*x + d)^n*c) + a^3)/(g*i^2*x
^3 + f*h^2 + (2*g*h*i + f*i^2)*x^2 + (g*h^2 + 2*f*h*i)*x), 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((a+b*ln(c*(e*x+d)**n))**3/(g*x+f)/(i*x+h)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}{\left (i x + h\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h)^2,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^3/((g*x + f)*(i*x + h)^2), x)