Integrand size = 21, antiderivative size = 174 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=-\frac {b n}{6 d^2 (d+e x)^2}-\frac {5 b n}{6 d^3 (d+e x)}-\frac {5 b n \log (x)}{6 d^4}+\frac {a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {11 b n \log (d+e x)}{6 d^4}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4} \]
-1/6*b*n/d^2/(e*x+d)^2-5/6*b*n/d^3/(e*x+d)-5/6*b*n*ln(x)/d^4+1/3*(a+b*ln(c *x^n))/d/(e*x+d)^3+1/2*(a+b*ln(c*x^n))/d^2/(e*x+d)^2-e*x*(a+b*ln(c*x^n))/d ^4/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))/d^4+11/6*b*n*ln(e*x+d)/d^4+b*n*poly log(2,-d/e/x)/d^4
Time = 0.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=\frac {\frac {3 a^2}{b n}+\frac {2 a d^3}{(d+e x)^3}+\frac {3 a d^2}{(d+e x)^2}-\frac {b d^2 n}{(d+e x)^2}+\frac {6 a d}{d+e x}-\frac {5 b d n}{d+e x}-11 b n \log (x)+\frac {6 a \log \left (c x^n\right )}{n}+\frac {2 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac {3 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {6 b d \log \left (c x^n\right )}{d+e x}+\frac {3 b \log ^2\left (c x^n\right )}{n}+11 b n \log (d+e x)-6 a \log \left (1+\frac {e x}{d}\right )-6 b \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-6 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 d^4} \]
((3*a^2)/(b*n) + (2*a*d^3)/(d + e*x)^3 + (3*a*d^2)/(d + e*x)^2 - (b*d^2*n) /(d + e*x)^2 + (6*a*d)/(d + e*x) - (5*b*d*n)/(d + e*x) - 11*b*n*Log[x] + ( 6*a*Log[c*x^n])/n + (2*b*d^3*Log[c*x^n])/(d + e*x)^3 + (3*b*d^2*Log[c*x^n] )/(d + e*x)^2 + (6*b*d*Log[c*x^n])/(d + e*x) + (3*b*Log[c*x^n]^2)/n + 11*b *n*Log[d + e*x] - 6*a*Log[1 + (e*x)/d] - 6*b*Log[c*x^n]*Log[1 + (e*x)/d] - 6*b*n*PolyLog[2, -((e*x)/d)])/(6*d^4)
Time = 1.01 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.47, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4}dx}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^3}dx}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^3 (d+e x)}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d (d+e x)^3}+\frac {1}{d^3 x}\right )dx}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^2}dx}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^2 (d+e x)}-\frac {e}{d (d+e x)^2}+\frac {1}{d^2 x}\right )dx}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \int \frac {1}{d+e x}dx}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}\) |
-((e*(-1/3*(a + b*Log[c*x^n])/(e*(d + e*x)^3) + (b*n*(1/(2*d*(d + e*x)^2) + 1/(d^2*(d + e*x)) + Log[x]/d^3 - Log[d + e*x]/d^3))/(3*e)))/d) + (-((e*( -1/2*(a + b*Log[c*x^n])/(e*(d + e*x)^2) + (b*n*(1/(d*(d + e*x)) + Log[x]/d ^2 - Log[d + e*x]/d^2))/(2*e)))/d) + (-((e*((x*(a + b*Log[c*x^n]))/(d*(d + e*x)) - (b*n*Log[d + e*x])/(d*e)))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c *x^n]))/d) + (b*n*PolyLog[2, -(d/(e*x))])/d)/d)/d)/d
3.1.59.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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] - Simp[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]
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] - Simp[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] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
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[(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.71 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.82
| method | result | size |
| risch | \(-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{4}}+\frac {b \ln \left (x^{n}\right )}{d^{3} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right )}{2 d^{2} \left (e x +d \right )^{2}}+\frac {b \ln \left (x^{n}\right )}{3 d \left (e x +d \right )^{3}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{4}}-\frac {5 b n}{6 d^{3} \left (e x +d \right )}-\frac {b n}{6 d^{2} \left (e x +d \right )^{2}}+\frac {11 b n \ln \left (e x +d \right )}{6 d^{4}}-\frac {11 b n \ln \left (x \right )}{6 d^{4}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{4}}+\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{4}}+\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{4}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (e x +d \right )}{d^{4}}+\frac {1}{d^{3} \left (e x +d \right )}+\frac {1}{2 d^{2} \left (e x +d \right )^{2}}+\frac {1}{3 d \left (e x +d \right )^{3}}+\frac {\ln \left (x \right )}{d^{4}}\right )\) | \(316\) |
-b*ln(x^n)/d^4*ln(e*x+d)+b*ln(x^n)/d^3/(e*x+d)+1/2*b*ln(x^n)/d^2/(e*x+d)^2 +1/3*b*ln(x^n)/d/(e*x+d)^3+b*ln(x^n)/d^4*ln(x)-5/6*b*n/d^3/(e*x+d)-1/6*b*n /d^2/(e*x+d)^2+11/6*b*n*ln(e*x+d)/d^4-11/6*b*n*ln(x)/d^4-1/2*b*n/d^4*ln(x) ^2+b*n/d^4*ln(e*x+d)*ln(-e*x/d)+b*n/d^4*dilog(-e*x/d)+(-1/2*I*b*Pi*csgn(I* c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b* Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(-1/d ^4*ln(e*x+d)+1/d^3/(e*x+d)+1/2/d^2/(e*x+d)^2+1/3/d/(e*x+d)^3+1/d^4*ln(x))
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x} \,d x } \]
integral((b*log(c*x^n) + a)/(e^4*x^5 + 4*d*e^3*x^4 + 6*d^2*e^2*x^3 + 4*d^3 *e*x^2 + d^4*x), x)
Time = 58.96 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.93 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=- \frac {a e \left (\begin {cases} \frac {x}{d^{4}} & \text {for}\: e = 0 \\- \frac {1}{3 e \left (d + e x\right )^{3}} & \text {otherwise} \end {cases}\right )}{d} - \frac {a e \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {a e \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{4}} + \frac {a \log {\left (x \right )}}{d^{4}} - \frac {b e^{3} n \left (\begin {cases} - \frac {1}{e^{4} x} & \text {for}\: d = 0 \\- \frac {3 d}{6 d^{2} e^{3} + 12 d e^{4} x + 6 e^{5} x^{2}} - \frac {4 e x}{6 d^{2} e^{3} + 12 d e^{4} x + 6 e^{5} x^{2}} - \frac {\log {\left (d + e x \right )}}{3 d e^{3}} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {b e^{3} \left (\begin {cases} \frac {1}{e^{4} x} & \text {for}\: d = 0 \\- \frac {1}{3 d \left (\frac {d}{x} + e\right )^{3}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} + \frac {3 b e^{2} n \left (\begin {cases} - \frac {1}{e^{3} x} & \text {for}\: d = 0 \\- \frac {1}{2 d e^{2} + 2 e^{3} x} - \frac {\log {\left (d + e x \right )}}{2 d e^{2}} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {3 b e^{2} \left (\begin {cases} \frac {1}{e^{3} x} & \text {for}\: d = 0 \\- \frac {1}{2 d \left (\frac {d}{x} + e\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} - \frac {3 b e n \left (\begin {cases} - \frac {1}{e^{2} x} & \text {for}\: d = 0 \\- \frac {\log {\left (d^{2} + d e x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {3 b e \left (\begin {cases} \frac {1}{e^{2} x} & \text {for}\: d = 0 \\- \frac {1}{\frac {d^{2}}{x} + d e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} + \frac {b n \left (\begin {cases} - \frac {1}{e x} & \text {for}\: d = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} + \operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + \operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {b \left (\begin {cases} \frac {1}{e x} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {d}{x} + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} \]
-a*e*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/d - a*e* Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/d**2 - a*e*Pi ecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d**3 - a*e*Piecewis e((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**4 + a*log(x)/d**4 - b*e**3*n *Piecewise((-1/(e**4*x), Eq(d, 0)), (-3*d/(6*d**2*e**3 + 12*d*e**4*x + 6*e **5*x**2) - 4*e*x/(6*d**2*e**3 + 12*d*e**4*x + 6*e**5*x**2) - log(d + e*x) /(3*d*e**3), True))/d**3 + b*e**3*Piecewise((1/(e**4*x), Eq(d, 0)), (-1/(3 *d*(d/x + e)**3), True))*log(c*x**n)/d**3 + 3*b*e**2*n*Piecewise((-1/(e**3 *x), Eq(d, 0)), (-1/(2*d*e**2 + 2*e**3*x) - log(d + e*x)/(2*d*e**2), True) )/d**3 - 3*b*e**2*Piecewise((1/(e**3*x), Eq(d, 0)), (-1/(2*d*(d/x + e)**2) , True))*log(c*x**n)/d**3 - 3*b*e*n*Piecewise((-1/(e**2*x), Eq(d, 0)), (-l og(d**2 + d*e*x)/(d*e), True))/d**3 + 3*b*e*Piecewise((1/(e**2*x), Eq(d, 0 )), (-1/(d**2/x + d*e), True))*log(c*x**n)/d**3 + b*n*Piecewise((-1/(e*x), Eq(d, 0)), (Piecewise((polylog(2, d*exp_polar(I*pi)/(e*x)), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(e)*log(x) + polylog(2, d*exp_polar(I*pi)/(e*x)), A bs(x) < 1), (-log(e)*log(1/x) + polylog(2, d*exp_polar(I*pi)/(e*x)), 1/Abs (x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1 ), ()), ((), (0, 0)), x)*log(e) + polylog(2, d*exp_polar(I*pi)/(e*x)), Tru e))/d, True))/d**3 - b*Piecewise((1/(e*x), Eq(d, 0)), (log(d/x + e)/d, Tru e))*log(c*x**n)/d**3
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x} \,d x } \]
1/6*a*((6*e^2*x^2 + 15*d*e*x + 11*d^2)/(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3*d^ 5*e*x + d^6) - 6*log(e*x + d)/d^4 + 6*log(x)/d^4) + b*integrate((log(c) + log(x^n))/(e^4*x^5 + 4*d*e^3*x^4 + 6*d^2*e^2*x^3 + 4*d^3*e*x^2 + d^4*x), x )
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^4} \,d x \]