3.59 \(\int \frac{(a+b \log (c (d+e x)^n))^3}{(f+g x)^4} \, dx\)

Optimal. Leaf size=564 \[ \frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,-\frac{e f-d g}{g (d+e x)}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}-\frac{b^3 e^3 n^3 \text{PolyLog}\left (2,-\frac{e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac{2 b^3 e^3 n^3 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac{2 b^3 e^3 n^3 \text{PolyLog}\left (3,-\frac{e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac{2 b^2 e^3 n^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac{b^2 e^3 n^2 \log \left (\frac{e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac{b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x) (e f-d g)^3}-\frac{b e^3 n \log \left (\frac{e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g (e f-d g)^3}-\frac{b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (e f-d g)^3}+\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2 (e f-d g)}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3} \]

[Out]

(b^2*e^2*n^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)^3*(f + g*x)) + (b*e*n*(a + b*Log[c*(d + e*x)^n
])^2)/(2*g*(e*f - d*g)*(f + g*x)^2) - (b*e^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/((e*f - d*g)^3*(f + g*x
)) - (a + b*Log[c*(d + e*x)^n])^3/(3*g*(f + g*x)^3) - (b^3*e^3*n^3*Log[f + g*x])/(g*(e*f - d*g)^3) + (2*b^2*e^
3*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)^3) + (b^2*e^3*n^2*(a + b*Log[c
*(d + e*x)^n])*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^3) - (b*e^3*n*(a + b*Log[c*(d + e*x)^n])^2*L
og[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^3) - (b^3*e^3*n^3*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/
(g*(e*f - d*g)^3) + (2*b^2*e^3*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*
f - d*g)^3) + (2*b^3*e^3*n^3*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)^3) + (2*b^3*e^3*n^3*Poly
Log[3, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^3)

________________________________________________________________________________________

Rubi [A]  time = 1.14029, antiderivative size = 525, normalized size of antiderivative = 0.93, number of steps used = 21, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ -\frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac{2 b^3 e^3 n^3 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac{3 b^2 e^3 n^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac{b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x) (e f-d g)^3}+\frac{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (e f-d g)^3}-\frac{b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^3}-\frac{b e^3 n \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g (e f-d g)^3}-\frac{b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (e f-d g)^3}+\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2 (e f-d g)}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^4,x]

[Out]

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

Rule 2398

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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 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 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 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 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}+\frac{(b e n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (f+g x)^3} \, dx}{g}\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^3} \, dx,x,d+e x\right )}{g}\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^3} \, dx,x,d+e x\right )}{e f-d g}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^2} \, dx,x,d+e x\right )}{g (e f-d g)}\\ &=\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^2} \, dx,x,d+e x\right )}{(e f-d g)^2}+\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )} \, dx,x,d+e x\right )}{g (e f-d g)^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^2} \, dx,x,d+e x\right )}{g (e f-d g)}\\ &=\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac{b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g)^3 (f+g x)}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\frac{e f-d g}{e}+\frac{g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^3}+\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x\right )}{g (e f-d g)^3}+\frac{\left (2 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\frac{e f-d g}{e}+\frac{g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^3}+\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^2} \, dx,x,d+e x\right )}{(e f-d g)^2}-\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )} \, dx,x,d+e x\right )}{g (e f-d g)^2}\\ &=\frac{b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^3 (f+g x)}+\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac{b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g)^3 (f+g x)}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}+\frac{2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g (e f-d g)^3}-\frac{b e^3 n \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 )}{g (e f-d g)^3}+\frac{e^3 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^3}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\frac{e f-d g}{e}+\frac{g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^3}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^3}+\frac{\left (2 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^3}-\frac{\left (b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{e f-d g}{e}+\frac{g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^3}-\frac{\left (2 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^3}\\ &=\frac{b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^3 (f+g x)}-\frac{b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^3}+\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac{b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g)^3 (f+g x)}+\frac{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (e f-d g)^3}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3}+\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g (e f-d g)^3}-\frac{b e^3 n \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 )}{g (e f-d g)^3}+\frac{2 b^3 e^3 n^3 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}-\frac{2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}-\frac{\left (b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^3}+\frac{\left (2 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^3}\\ &=\frac{b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^3 (f+g x)}-\frac{b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^3}+\frac{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac{b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g)^3 (f+g x)}+\frac{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (e f-d g)^3}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac{b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3}+\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g (e f-d g)^3}-\frac{b e^3 n \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 )}{g (e f-d g)^3}+\frac{3 b^3 e^3 n^3 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}-\frac{2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac{2 b^3 e^3 n^3 \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}\\ \end{align*}

Mathematica [A]  time = 1.29668, size = 843, normalized size = 1.49 \[ \frac{-2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 (e f-d g)^3-6 b n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 (e f-d g)^3+3 b e n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 (e f-d g)^2+6 b e^2 n (f+g x)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 (e f-d g)+6 b e^3 n (f+g x)^3 \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-6 b e^3 n (f+g x)^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (3 e^3 \log \left (\frac{e (f+g x)}{e f-d g}\right ) (f+g x)^3-2 e^3 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) (f+g x)^3+e^2 g (d+e x) (f+g x)^2+e \log (d+e x) \left (g^2 (d+e x)^2-4 e g (f+g x) (d+e x)-2 e^2 (f+g x)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) (f+g x)+g \left (g^2 d^3-3 e f g d^2+3 e^2 f^2 d+e^3 x \left (3 f^2+3 g x f+g^2 x^2\right )\right ) \log ^2(d+e x)\right )+b^3 n^3 \left (-6 e^3 \log \left (\frac{e (f+g x)}{e f-d g}\right ) (f+g x)^3+18 e^3 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) (f+g x)^3+12 e^3 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right ) (f+g x)^3+6 e^2 \log (d+e x) \left (g (d+e x)+3 e (f+g x) \log \left (\frac{e (f+g x)}{e f-d g}\right )-2 e (f+g x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )\right ) (f+g x)^2+3 e \log ^2(d+e x) \left (g^2 (d+e x)^2-4 e g (f+g x) (d+e x)-2 e^2 (f+g x)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) (f+g x)+2 g \left (g^2 d^3-3 e f g d^2+3 e^2 f^2 d+e^3 x \left (3 f^2+3 g x f+g^2 x^2\right )\right ) \log ^3(d+e x)\right )}{6 g (e f-d g)^3 (f+g x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^4,x]

[Out]

(3*b*e*(e*f - d*g)^2*n*(f + g*x)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 6*b*e^2*(e*f - d*g)*n*(f +
g*x)^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 6*b*(e*f - d*g)^3*n*Log[d + e*x]*(a - b*n*Log[d + e*x
] + b*Log[c*(d + e*x)^n])^2 + 6*b*e^3*n*(f + g*x)^3*Log[d + e*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])
^2 - 2*(e*f - d*g)^3*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3 - 6*b*e^3*n*(f + g*x)^3*(a - b*n*Log[d +
e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x] + 6*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e^2*g*(
d + e*x)*(f + g*x)^2 + g*(3*d*e^2*f^2 - 3*d^2*e*f*g + d^3*g^2 + e^3*x*(3*f^2 + 3*f*g*x + g^2*x^2))*Log[d + e*x
]^2 + 3*e^3*(f + g*x)^3*Log[(e*(f + g*x))/(e*f - d*g)] + e*(f + g*x)*Log[d + e*x]*(g^2*(d + e*x)^2 - 4*e*g*(d
+ e*x)*(f + g*x) - 2*e^2*(f + g*x)^2*Log[(e*(f + g*x))/(e*f - d*g)]) - 2*e^3*(f + g*x)^3*PolyLog[2, (g*(d + e*
x))/(-(e*f) + d*g)]) + b^3*n^3*(2*g*(3*d*e^2*f^2 - 3*d^2*e*f*g + d^3*g^2 + e^3*x*(3*f^2 + 3*f*g*x + g^2*x^2))*
Log[d + e*x]^3 - 6*e^3*(f + g*x)^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*e*(f + g*x)*Log[d + e*x]^2*(g^2*(d + e*x
)^2 - 4*e*g*(d + e*x)*(f + g*x) - 2*e^2*(f + g*x)^2*Log[(e*(f + g*x))/(e*f - d*g)]) + 18*e^3*(f + g*x)^3*PolyL
og[2, (g*(d + e*x))/(-(e*f) + d*g)] + 6*e^2*(f + g*x)^2*Log[d + e*x]*(g*(d + e*x) + 3*e*(f + g*x)*Log[(e*(f +
g*x))/(e*f - d*g)] - 2*e*(f + g*x)*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + 12*e^3*(f + g*x)^3*PolyLog[3, (
g*(d + e*x))/(-(e*f) + d*g)]))/(6*g*(e*f - d*g)^3*(f + g*x)^3)

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Maple [F]  time = 2.224, 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 ) ^{4}}}\, 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)^4,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (\frac{2 \, e^{2} \log \left (e x + d\right )}{e^{3} f^{3} g - 3 \, d e^{2} f^{2} g^{2} + 3 \, d^{2} e f g^{3} - d^{3} g^{4}} - \frac{2 \, e^{2} \log \left (g x + f\right )}{e^{3} f^{3} g - 3 \, d e^{2} f^{2} g^{2} + 3 \, d^{2} e f g^{3} - d^{3} g^{4}} + \frac{2 \, e g x + 3 \, e f - d g}{e^{2} f^{4} g - 2 \, d e f^{3} g^{2} + d^{2} f^{2} g^{3} +{\left (e^{2} f^{2} g^{3} - 2 \, d e f g^{4} + d^{2} g^{5}\right )} x^{2} + 2 \,{\left (e^{2} f^{3} g^{2} - 2 \, d e f^{2} g^{3} + d^{2} f g^{4}\right )} x}\right )} a^{2} b e n - \frac{b^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{3}}{3 \,{\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} - \frac{a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right )}{g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g} - \frac{a^{3}}{3 \,{\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} + \int \frac{b^{3} d g \log \left (c\right )^{3} + 3 \, a b^{2} d g \log \left (c\right )^{2} +{\left (3 \, a b^{2} d g +{\left (e f n + 3 \, d g \log \left (c\right )\right )} b^{3} +{\left (3 \, a b^{2} e g +{\left (e g n + 3 \, e g \log \left (c\right )\right )} b^{3}\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} +{\left (b^{3} e g \log \left (c\right )^{3} + 3 \, a b^{2} e g \log \left (c\right )^{2}\right )} x + 3 \,{\left (b^{3} d g \log \left (c\right )^{2} + 2 \, a b^{2} d g \log \left (c\right ) +{\left (b^{3} e g \log \left (c\right )^{2} + 2 \, a b^{2} e g \log \left (c\right )\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{e g^{5} x^{5} + d f^{4} g +{\left (4 \, e f g^{4} + d g^{5}\right )} x^{4} + 2 \,{\left (3 \, e f^{2} g^{3} + 2 \, d f g^{4}\right )} x^{3} + 2 \,{\left (2 \, e f^{3} g^{2} + 3 \, d f^{2} g^{3}\right )} x^{2} +{\left (e f^{4} g + 4 \, d f^{3} g^{2}\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)^4,x, algorithm="maxima")

[Out]

1/2*(2*e^2*log(e*x + d)/(e^3*f^3*g - 3*d*e^2*f^2*g^2 + 3*d^2*e*f*g^3 - d^3*g^4) - 2*e^2*log(g*x + f)/(e^3*f^3*
g - 3*d*e^2*f^2*g^2 + 3*d^2*e*f*g^3 - d^3*g^4) + (2*e*g*x + 3*e*f - d*g)/(e^2*f^4*g - 2*d*e*f^3*g^2 + d^2*f^2*
g^3 + (e^2*f^2*g^3 - 2*d*e*f*g^4 + d^2*g^5)*x^2 + 2*(e^2*f^3*g^2 - 2*d*e*f^2*g^3 + d^2*f*g^4)*x))*a^2*b*e*n -
1/3*b^3*log((e*x + d)^n)^3/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g) - a^2*b*log((e*x + d)^n*c)/(g^4*x^3 +
3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g) - 1/3*a^3/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g) + integrate((b^3*d
*g*log(c)^3 + 3*a*b^2*d*g*log(c)^2 + (3*a*b^2*d*g + (e*f*n + 3*d*g*log(c))*b^3 + (3*a*b^2*e*g + (e*g*n + 3*e*g
*log(c))*b^3)*x)*log((e*x + d)^n)^2 + (b^3*e*g*log(c)^3 + 3*a*b^2*e*g*log(c)^2)*x + 3*(b^3*d*g*log(c)^2 + 2*a*
b^2*d*g*log(c) + (b^3*e*g*log(c)^2 + 2*a*b^2*e*g*log(c))*x)*log((e*x + d)^n))/(e*g^5*x^5 + d*f^4*g + (4*e*f*g^
4 + d*g^5)*x^4 + 2*(3*e*f^2*g^3 + 2*d*f*g^4)*x^3 + 2*(2*e*f^3*g^2 + 3*d*f^2*g^3)*x^2 + (e*f^4*g + 4*d*f^3*g^2)
*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^{4} x^{4} + 4 \, f g^{3} x^{3} + 6 \, f^{2} g^{2} x^{2} + 4 \, f^{3} g x + f^{4}}, 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)^4,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^4*x^4
+ 4*f*g^3*x^3 + 6*f^2*g^2*x^2 + 4*f^3*g*x + f^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((a+b*ln(c*(e*x+d)**n))**3/(g*x+f)**4,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 )}^{4}}\,{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)^4,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^3/(g*x + f)^4, x)