Integrand size = 23, antiderivative size = 217 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^2}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d^2} \]
-e*x*(a+b*ln(c*x^n))^3/d^2/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))^3/d^2+3*b*n *(a+b*ln(c*x^n))^2*ln(1+e*x/d)/d^2+3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-d/e/ x)/d^2+6*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/d^2+6*b^2*n^2*(a+b*ln(c *x^n))*polylog(3,-d/e/x)/d^2-6*b^3*n^3*polylog(3,-e*x/d)/d^2+6*b^3*n^3*pol ylog(4,-d/e/x)/d^2
Time = 0.36 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\frac {4 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+4 (d+e x) \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3-4 (d+e x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \log (d+e x)+6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (-2 e x \log (x)+(d+e x) \log ^2(x)+2 (d+e x) \log (d+e x)-2 (d+e x) \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )+4 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\log (x) \left ((d+e x) \log ^2(x)+6 (d+e x) \log \left (1+\frac {e x}{d}\right )-3 \log (x) \left (e x+(d+e x) \log \left (1+\frac {e x}{d}\right )\right )\right )-6 (d+e x) (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )+b^3 n^3 \left ((d+e x) \log ^4(x)-4 \left (\log ^2(x) \left (e x \log (x)-3 (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-6 (d+e x) \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-4 (d+e x) \left (\log ^3(x) \log \left (1+\frac {e x}{d}\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 \log (x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+6 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )\right )\right )}{4 d^2 (d+e x)} \]
(4*d*(a - b*n*Log[x] + b*Log[c*x^n])^3 + 4*(d + e*x)*Log[x]*(a - b*n*Log[x ] + b*Log[c*x^n])^3 - 4*(d + e*x)*(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*(-2*e*x*Log[x] + (d + e*x )*Log[x]^2 + 2*(d + e*x)*Log[d + e*x] - 2*(d + e*x)*(Log[x]*Log[1 + (e*x)/ d] + PolyLog[2, -((e*x)/d)])) + 4*b^2*n^2*(a - b*n*Log[x] + b*Log[c*x^n])* (Log[x]*((d + e*x)*Log[x]^2 + 6*(d + e*x)*Log[1 + (e*x)/d] - 3*Log[x]*(e*x + (d + e*x)*Log[1 + (e*x)/d])) - 6*(d + e*x)*(-1 + Log[x])*PolyLog[2, -(( e*x)/d)] + 6*(d + e*x)*PolyLog[3, -((e*x)/d)]) + b^3*n^3*((d + e*x)*Log[x] ^4 - 4*(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)]) - 4*( d + e*x)*(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^2*(d + e*x))
Time = 1.01 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2789, 2755, 2754, 2779, 2821, 2830, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2}dx}{d}\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {3 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{x}dx-\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (-\left (\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )\right )-b n \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}\) |
-((e*((x*(a + b*Log[c*x^n])^3)/(d*(d + e*x)) - (3*b*n*(((a + b*Log[c*x^n]) ^2*Log[1 + (e*x)/d])/e - (2*b*n*(-((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d )]) + b*n*PolyLog[3, -((e*x)/d)]))/e))/d))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d) + (3*b*n*((a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))] - 2*b*n*(-((a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))]) - b*n*PolyLog[4, -(d /(e*x))])))/d)/d
3.2.22.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[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]
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] + Simp[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]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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] + Simp[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]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.55 (sec) , antiderivative size = 1373, normalized size of antiderivative = 6.33
6*b^3/d^2*ln(e*x+d)*ln(-e*x/d)*ln(x)*n^3-6*b^3/d^2*n^2*ln(x^n)*ln(e*x+d)*l n(-e*x/d)+3*b^3/d^2*ln(e*x+d)*ln(-e*x/d)*ln(x)^2*n^3-6*b^3/d^2*ln(x^n)*dil og(-e*x/d)*ln(x)*n^2+3*b^3/d^2*n^2*ln(x^n)*ln(e*x+d)*ln(x)^2-6*b^3/d^2*ln( x^n)*ln(e*x+d)*ln(-e*x/d)*ln(x)*n^2+3/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csg n(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^2*ln(e*x+d)+l n(x^n)^2/d/(e*x+d)+ln(x^n)^2/d^2*ln(x)-2*n*(-ln(x^n)/d^2*ln(e*x+d)+ln(x^n) /d^2*ln(x)-1/2/d^2*n*ln(x)^2+1/d^2*n*ln(e*x+d)*ln(-e*x/d)+1/d^2*n*dilog(-e *x/d)+1/2/d^2*ln(x^n)*ln(x)^2-1/6/d^2*ln(x)^3*n-1/d^2*((ln(x^n)-n*ln(x))*( dilog(-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*l n(1+e*x/d)-ln(x)*polylog(2,-e*x/d)+polylog(3,-e*x/d)))))-2*b^3/d^2*n^3*ln( e*x+d)*ln(x)^3+2*b^3/d^2*n^3*ln(x)^3*ln(1+e*x/d)+3*b^3*n/d^2*ln(x^n)^2*dil og(-e*x/d)+3*b^3*n*ln(x^n)^2/d^2*ln(e*x+d)-3*b^3*n*ln(x^n)^2/d^2*ln(x)+3*b ^3/d^2*n^3*ln(x)^2*polylog(2,-e*x/d)+6*b^3/d^2*n^2*ln(x^n)*polylog(3,-e*x/ d)+b^3/d^2*n^2*ln(x^n)*ln(x)^3-3/2*b^3*n/d^2*ln(x^n)^2*ln(x)^2+3*b^3/d^2*n ^2*ln(x^n)*ln(x)^2+6*b^3/d^2*dilog(-e*x/d)*ln(x)*n^3-6*b^3/d^2*n^2*ln(x^n) *dilog(-e*x/d)-3*b^3/d^2*n^3*ln(e*x+d)*ln(x)^2+3*b^3/d^2*n^3*ln(x)^2*ln(1+ e*x/d)+6*b^3/d^2*n^3*ln(x)*polylog(2,-e*x/d)+3*b^3/d^2*dilog(-e*x/d)*ln(x) ^2*n^3+3/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*c sgn(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)^...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2} x} \,d x } \]
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^2*x^3 + 2*d*e*x^2 + d^2*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{x \left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2} x} \,d x } \]
a^3*(1/(d*e*x + d^2) - log(e*x + d)/d^2 + log(x)/d^2) + integrate((b^3*log (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^2*x^3 + 2*d*e*x^2 + d^2*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x\,{\left (d+e\,x\right )}^2} \,d x \]