Integrand size = 21, antiderivative size = 80 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {b n \log (d+e x)}{d^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2} \]
-e*x*(a+b*ln(c*x^n))/d^2/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))/d^2+b*n*ln(e* x+d)/d^2+b*n*polylog(2,-d/e/x)/d^2
Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\frac {\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 b n (\log (x)-\log (d+e x))-2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^2} \]
((2*d*(a + b*Log[c*x^n]))/(d + e*x) + (a + b*Log[c*x^n])^2/(b*n) - 2*b*n*( Log[x] - Log[d + e*x]) - 2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 2*b*n*Pol yLog[2, -((e*x)/d)])/(2*d^2)
Time = 0.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2}dx}{d}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \int \frac {1}{d+e x}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\) |
-((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
3.1.43.3.1 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_.) + 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_.)/((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 = 0.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.86
| method | result | size |
| risch | \(-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}+\frac {b \ln \left (x^{n}\right )}{d \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{2}}+\frac {b n \ln \left (e x +d \right )}{d^{2}}-\frac {b n \ln \left (x \right )}{d^{2}}+\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}+\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {1}{d \left (e x +d \right )}+\frac {\ln \left (x \right )}{d^{2}}\right )\) | \(229\) |
-b*ln(x^n)/d^2*ln(e*x+d)+b*ln(x^n)/d/(e*x+d)+b*ln(x^n)/d^2*ln(x)-1/2*b*n/d ^2*ln(x)^2+b*n*ln(e*x+d)/d^2-b*n/d^2*ln(x)+b*n/d^2*ln(e*x+d)*ln(-e*x/d)+b* n/d^2*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^2*ln(e*x+d)+1/d/(e*x+d)+1/d^2*ln( x))
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{2} x} \,d x } \]
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x \left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{2} x} \,d x } \]
a*(1/(d*e*x + d^2) - log(e*x + d)/d^2 + log(x)/d^2) + b*integrate((log(c) + log(x^n))/(e^2*x^3 + 2*d*e*x^2 + d^2*x), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^2} \,d x \]