\(\int \frac {(a+b \log (c x^n))^3}{x (d+e x)^3} \, dx\) [123]

Optimal result

Integrand size = 23, antiderivative size = 361 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\frac {3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {9 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}+\frac {9 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {9 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^3}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^3}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )}{d^3} \]

[Out]

3/2*b*e*n*x*(a+b*ln(c*x^n))^2/d^3/(e*x+d)-1/2*(a+b*ln(c*x^n))^3/d^3+1/2*(a+b*ln(c*x^n))^3/d/(e*x+d)^2-e*x*(a+b
*ln(c*x^n))^3/d^3/(e*x+d)+1/4*(a+b*ln(c*x^n))^4/b/d^3/n-3*b^2*n^2*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^3+9/2*b*n*(a+b
*ln(c*x^n))^2*ln(1+e*x/d)/d^3-(a+b*ln(c*x^n))^3*ln(1+e*x/d)/d^3-3*b^3*n^3*polylog(2,-e*x/d)/d^3+9*b^2*n^2*(a+b
*ln(c*x^n))*polylog(2,-e*x/d)/d^3-3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-e*x/d)/d^3-9*b^3*n^3*polylog(3,-e*x/d)/d^
3+6*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-e*x/d)/d^3-6*b^3*n^3*polylog(4,-e*x/d)/d^3

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.13, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2389, 2379, 2421, 2430, 6724, 2355, 2354, 2356, 2438} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac {3 b^2 n^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}+\frac {3 b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac {3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac {3 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^3}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d^3} \]

[In]

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

[Out]

(3*b*e*n*x*(a + b*Log[c*x^n])^2)/(2*d^3*(d + e*x)) + (3*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/(2*d^3) + (
a + b*Log[c*x^n])^3/(2*d*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^3)/(d^3*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*
Log[c*x^n])^3)/d^3 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 +
(e*x)/d])/d^3 - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^3 + (3*b*n*(a + b*Log[c*x^n])^2*PolyL
og[2, -(d/(e*x))])/d^3 - (3*b^3*n^3*PolyLog[2, -((e*x)/d)])/d^3 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((
e*x)/d)])/d^3 - (3*b^3*n^3*PolyLog[3, -(d/(e*x))])/d^3 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))])
/d^3 - (6*b^3*n^3*PolyLog[3, -((e*x)/d)])/d^3 + (6*b^3*n^3*PolyLog[4, -(d/(e*x))])/d^3

Rule 2354

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 2355

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 2356

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 2379

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

Rule 2389

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 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 6724

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx}{d} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx}{d^2}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2} \, dx}{d^2}-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{2 d} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}+\frac {(3 b n) \int \frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^3}-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{2 d^2}+\frac {(3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^3}+\frac {(3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{2 d^2} \\ & = \frac {3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac {3 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}-\frac {\left (3 b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^3}-\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^3}-\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{x} \, dx}{d^3}-\frac {\left (3 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3} \\ & = \frac {3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac {3 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^3}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {d}{e x}\right )}{d^3}+\frac {\left (3 b^3 n^3\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^3}+\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {d}{e x}\right )}{x} \, dx}{d^3}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^3}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {d}{e x}\right )}{x} \, dx}{d^3} \\ & = \frac {3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac {3 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}-\frac {3 b^3 n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^3}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^3}-\frac {3 b^3 n^3 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^3}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {d}{e x}\right )}{d^3}-\frac {6 b^3 n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^3}+\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {d}{e x}\right )}{d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\frac {2 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+4 d (d+e x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+4 (d+e x)^2 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3-4 (d+e x)^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \log (d+e x)+6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left ((d+e x)^2 \log ^2(x)+(d+e x) (-d+3 (d+e x) \log (d+e x))-\log (x) \left (e x (4 d+3 e x)+2 (d+e x)^2 \log \left (1+\frac {e x}{d}\right )\right )-2 (d+e x)^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+2 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (-3 e x (2 d+e x) \log ^2(x)+2 (d+e x)^2 \log ^3(x)-6 (d+e x)^2 \log (d+e x)+6 (d+e x) \log (x) \left (e x+(d+e x) \log \left (1+\frac {e x}{d}\right )\right )+6 (d+e x)^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 (d+e x) \left (\log (x) \left (e x \log (x)-2 (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-2 (d+e x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-6 (d+e x)^2 \left (\log ^2(x) \log \left (1+\frac {e x}{d}\right )+2 \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )\right )+b^3 n^3 \left ((d+e x)^2 \log ^4(x)-4 (d+e x) \left (\log ^2(x) \left (e x \log (x)-3 (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-6 (d+e x) \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-2 \left (\log (x) \left (e x (2 d+e x) \log ^2(x)+6 (d+e x)^2 \log \left (1+\frac {e x}{d}\right )-3 (d+e x) \log (x) \left (e x+(d+e x) \log \left (1+\frac {e x}{d}\right )\right )\right )-6 (d+e x)^2 (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x)^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-4 (d+e x)^2 \left (\log ^3(x) \log \left (1+\frac {e x}{d}\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 \log (x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+6 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )\right )\right )}{4 d^3 (d+e x)^2} \]

[In]

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

[Out]

(2*d^2*(a - b*n*Log[x] + b*Log[c*x^n])^3 + 4*d*(d + e*x)*(a - b*n*Log[x] + b*Log[c*x^n])^3 + 4*(d + e*x)^2*Log
[x]*(a - b*n*Log[x] + b*Log[c*x^n])^3 - 4*(d + e*x)^2*(a - b*n*Log[x] + b*Log[c*x^n])^3*Log[d + e*x] + 6*b*n*(
a - b*n*Log[x] + b*Log[c*x^n])^2*((d + e*x)^2*Log[x]^2 + (d + e*x)*(-d + 3*(d + e*x)*Log[d + e*x]) - Log[x]*(e
*x*(4*d + 3*e*x) + 2*(d + e*x)^2*Log[1 + (e*x)/d]) - 2*(d + e*x)^2*PolyLog[2, -((e*x)/d)]) + 2*b^2*n^2*(a - b*
n*Log[x] + b*Log[c*x^n])*(-3*e*x*(2*d + e*x)*Log[x]^2 + 2*(d + e*x)^2*Log[x]^3 - 6*(d + e*x)^2*Log[d + e*x] +
6*(d + e*x)*Log[x]*(e*x + (d + e*x)*Log[1 + (e*x)/d]) + 6*(d + e*x)^2*PolyLog[2, -((e*x)/d)] - 6*(d + e*x)*(Lo
g[x]*(e*x*Log[x] - 2*(d + e*x)*Log[1 + (e*x)/d]) - 2*(d + e*x)*PolyLog[2, -((e*x)/d)]) - 6*(d + e*x)^2*(Log[x]
^2*Log[1 + (e*x)/d] + 2*Log[x]*PolyLog[2, -((e*x)/d)] - 2*PolyLog[3, -((e*x)/d)])) + b^3*n^3*((d + e*x)^2*Log[
x]^4 - 4*(d + e*x)*(Log[x]^2*(e*x*Log[x] - 3*(d + e*x)*Log[1 + (e*x)/d]) - 6*(d + e*x)*Log[x]*PolyLog[2, -((e*
x)/d)] + 6*(d + e*x)*PolyLog[3, -((e*x)/d)]) - 2*(Log[x]*(e*x*(2*d + e*x)*Log[x]^2 + 6*(d + e*x)^2*Log[1 + (e*
x)/d] - 3*(d + e*x)*Log[x]*(e*x + (d + e*x)*Log[1 + (e*x)/d])) - 6*(d + e*x)^2*(-1 + Log[x])*PolyLog[2, -((e*x
)/d)] + 6*(d + e*x)^2*PolyLog[3, -((e*x)/d)]) - 4*(d + e*x)^2*(Log[x]^3*Log[1 + (e*x)/d] + 3*Log[x]^2*PolyLog[
2, -((e*x)/d)] - 6*Log[x]*PolyLog[3, -((e*x)/d)] + 6*PolyLog[4, -((e*x)/d)])))/(4*d^3*(d + e*x)^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.63 (sec) , antiderivative size = 1607, normalized size of antiderivative = 4.45

method	result	size
risch	\(\text {Expression too large to display}\)	\(1607\)

[In]

int((a+b*ln(c*x^n))^3/x/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

-6*b^3/d^3*ln(x)*ln(x^n)*ln(e*x+d)*ln(-e*x/d)*n^2+3/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn
(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*b*(-ln(x^n)/d
^3*ln(e*x+d)+ln(x^n)/d^2/(e*x+d)+1/2*ln(x^n)/d/(e*x+d)^2+ln(x^n)/d^3*ln(x)-1/2*n*(1/d^2/(e*x+d)-3/d^3*ln(e*x+d
)+3/d^3*ln(x)+1/d^3*ln(x)^2-2/d^3*ln(e*x+d)*ln(-e*x/d)-2/d^3*dilog(-e*x/d)))+3/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n
)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b
*ln(c)+2*a)*b^2*(-ln(x^n)^2/d^3*ln(e*x+d)+ln(x^n)^2/d^2/(e*x+d)+1/2*ln(x^n)^2/d/(e*x+d)^2+ln(x^n)^2/d^3*ln(x)-
n*(1/d^2*(ln(x^n)/(e*x+d)-3*ln(x^n)/d*ln(e*x+d)+3*ln(x^n)/d*ln(x)-n*(-1/d*ln(e*x+d)+1/d*ln(x)+3/2/d*ln(x)^2-3/
d*ln(e*x+d)*ln(-e*x/d)-3/d*dilog(-e*x/d)))+1/d^3*ln(x^n)*ln(x)^2-1/3/d^3*ln(x)^3*n-2/d^3*((ln(x^n)-n*ln(x))*(d
ilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))+n*(1/2*ln(e*x+d)*ln(x)^2-1/2*ln(x)^2*ln(1+e*x/d)-ln(x)*polylog(2,-e*x/d)+po
lylog(3,-e*x/d)))))+1/8*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^3*(-1/d^3*ln(e*x+d)+1/d^2/(e*x+d)+1/2/d/(e*x+
d)^2+1/d^3*ln(x))-9/2*b^3/d^3*n^3*ln(e*x+d)*ln(x)^2-3/2*b^3/d^3*n^3*ln(x)^2+3*b^3/d^3*n^3*dilog(-e*x/d)-3/2*b^
3/d^3*ln(x)^3*n^3-1/4*b^3/d^3*ln(x)^4*n^3-b^3*ln(x^n)^3/d^3*ln(e*x+d)+b^3*ln(x^n)^3/d^2/(e*x+d)+1/2*b^3*ln(x^n
)^3/d/(e*x+d)^2+b^3*ln(x^n)^3/d^3*ln(x)+9*b^3/d^3*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^3-9*b^3/d^3*n^2*ln(x^n)*ln(e*x+
d)*ln(-e*x/d)+3*b^3/d^3*ln(x)^2*ln(e*x+d)*ln(-e*x/d)*n^3-6*b^3/d^3*ln(x)*ln(x^n)*dilog(-e*x/d)*n^2+3*b^3/d^3*n
^2*ln(x)^2*ln(x^n)*ln(e*x+d)-3*b^3/d^3*n^2*ln(x)^2*ln(x^n)*ln(1+e*x/d)-6*b^3/d^3*n^2*ln(x)*ln(x^n)*polylog(2,-
e*x/d)+3*b^3*n/d^3*ln(x^n)^2*ln(e*x+d)*ln(-e*x/d)+9/2*b^3/d^3*n^3*ln(x)^2*ln(1+e*x/d)+9*b^3/d^3*n^3*ln(x)*poly
log(2,-e*x/d)+9/2*b^3/d^3*n^2*ln(x^n)*ln(x)^2-3/2*b^3*n/d^3*ln(x^n)^2*ln(x)^2+9/2*b^3*n*ln(x^n)^2/d^3*ln(e*x+d
)-3/2*b^3*n*ln(x^n)^2/d^2/(e*x+d)-9/2*b^3*n*ln(x^n)^2/d^3*ln(x)+3*b^3/d^3*ln(x)^2*dilog(-e*x/d)*n^3-2*b^3/d^3*
n^3*ln(e*x+d)*ln(x)^3+2*b^3/d^3*n^3*ln(x)^3*ln(1+e*x/d)+3*b^3/d^3*n^3*ln(x)^2*polylog(2,-e*x/d)+6*b^3/d^3*n^2*
ln(x^n)*polylog(3,-e*x/d)+b^3/d^3*n^2*ln(x^n)*ln(x)^3+3*b^3*n/d^3*ln(x^n)^2*dilog(-e*x/d)-3*b^3/d^3*n^2*ln(x^n
)*ln(e*x+d)+3*b^3/d^3*n^2*ln(x^n)*ln(x)+3*b^3/d^3*n^3*ln(e*x+d)*ln(-e*x/d)+9*b^3/d^3*ln(x)*dilog(-e*x/d)*n^3-9
*b^3/d^3*n^2*ln(x^n)*dilog(-e*x/d)-9*b^3*n^3*polylog(3,-e*x/d)/d^3-6*b^3*n^3*polylog(4,-e*x/d)/d^3

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3} x} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{x \left (d + e x\right )^{3}}\, dx \]

[In]

integrate((a+b*ln(c*x**n))**3/x/(e*x+d)**3,x)

[Out]

Integral((a + b*log(c*x**n))**3/(x*(d + e*x)**3), x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3} x} \,d x } \]

[In]

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

[Out]

1/2*a^3*((2*e*x + 3*d)/(d^2*e^2*x^2 + 2*d^3*e*x + d^4) - 2*log(e*x + d)/d^3 + 2*log(x)/d^3) + integrate((b^3*l
og(c)^3 + b^3*log(x^n)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + 3*(b^3*log(c) + a*b^2)*log(x^n)^2 + 3*(b^3*log(
c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x^n))/(e^3*x^4 + 3*d*e^2*x^3 + 3*d^2*e*x^2 + d^3*x), x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3} x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x\,{\left (d+e\,x\right )}^3} \,d x \]

[In]

int((a + b*log(c*x^n))^3/(x*(d + e*x)^3),x)

[Out]

int((a + b*log(c*x^n))^3/(x*(d + e*x)^3), x)