Integrand size = 23, antiderivative size = 169 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac {b n \log (x)}{2 d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^2} \]
-1/2*b*n/d^2/r^2/(d+e*x^r)-1/2*b*n*ln(x)/d^3/r+1/2*(a+b*ln(c*x^n))/d/r/(d+ e*x^r)^2-e*x^r*(a+b*ln(c*x^n))/d^3/r/(d+e*x^r)-(a+b*ln(c*x^n))*ln(1+d/e/(x ^r))/d^3/r+3/2*b*n*ln(d+e*x^r)/d^3/r^2+b*n*polylog(2,-d/e/(x^r))/d^3/r^2
Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\frac {\frac {d^2 r \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2}+\frac {d \left (-b n+2 a r+2 b r \log \left (c x^n\right )\right )}{d+e x^r}+3 b n \log \left (d-d x^r\right )-2 a r \log \left (d-d x^r\right )+2 b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b n \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )}{2 d^3 r^2} \]
((d^2*r*(a + b*Log[c*x^n]))/(d + e*x^r)^2 + (d*(-(b*n) + 2*a*r + 2*b*r*Log [c*x^n]))/(d + e*x^r) + 3*b*n*Log[d - d*x^r] - 2*a*r*Log[d - d*x^r] + 2*b* r*(n*Log[x] - Log[c*x^n])*Log[d - d*x^r] + 2*b*n*((r^2*Log[x]^2)/2 + (-(r* Log[x]) + Log[-((e*x^r)/d)])*Log[d + e*x^r] + PolyLog[2, 1 + (e*x^r)/d]))/ (2*d^3*r^2)
Time = 0.97 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2791, 2776, 798, 54, 2009, 2791, 2773, 792, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx\) |
| \(\Big \downarrow \) 2791 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )^2}dx}{d}-\frac {e \int \frac {x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{\left (e x^r+d\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 2776 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x \left (e x^r+d\right )^2}dx}{2 e r}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {x^{-r}}{\left (e x^r+d\right )^2}dx^r}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \int \left (\frac {x^{-r}}{d^2}-\frac {e}{d^2 \left (e x^r+d\right )}-\frac {e}{d \left (e x^r+d\right )^2}\right )dx^r}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )^2}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^r\right )}{d^2}+\frac {\log \left (x^r\right )}{d^2}+\frac {1}{d \left (d+e x^r\right )}\right )}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2791 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )}dx}{d}-\frac {e \int \frac {x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{\left (e x^r+d\right )^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^r\right )}{d^2}+\frac {\log \left (x^r\right )}{d^2}+\frac {1}{d \left (d+e x^r\right )}\right )}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )}dx}{d}-\frac {e \left (\frac {x^r \left (a+b \log \left (c x^n\right )\right )}{d r \left (d+e x^r\right )}-\frac {b n \int \frac {x^{r-1}}{e x^r+d}dx}{d r}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^r\right )}{d^2}+\frac {\log \left (x^r\right )}{d^2}+\frac {1}{d \left (d+e x^r\right )}\right )}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^r+d\right )}dx}{d}-\frac {e \left (\frac {x^r \left (a+b \log \left (c x^n\right )\right )}{d r \left (d+e x^r\right )}-\frac {b n \log \left (d+e x^r\right )}{d e r^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^r\right )}{d^2}+\frac {\log \left (x^r\right )}{d^2}+\frac {1}{d \left (d+e x^r\right )}\right )}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {\frac {b n \int \frac {\log \left (\frac {d x^{-r}}{e}+1\right )}{x}dx}{d r}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r}}{d}-\frac {e \left (\frac {x^r \left (a+b \log \left (c x^n\right )\right )}{d r \left (d+e x^r\right )}-\frac {b n \log \left (d+e x^r\right )}{d e r^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^r\right )}{d^2}+\frac {\log \left (x^r\right )}{d^2}+\frac {1}{d \left (d+e x^r\right )}\right )}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r}}{d}-\frac {e \left (\frac {x^r \left (a+b \log \left (c x^n\right )\right )}{d r \left (d+e x^r\right )}-\frac {b n \log \left (d+e x^r\right )}{d e r^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log \left (d+e x^r\right )}{d^2}+\frac {\log \left (x^r\right )}{d^2}+\frac {1}{d \left (d+e x^r\right )}\right )}{2 e r^2}-\frac {a+b \log \left (c x^n\right )}{2 e r \left (d+e x^r\right )^2}\right )}{d}\) |
-((e*(-1/2*(a + b*Log[c*x^n])/(e*r*(d + e*x^r)^2) + (b*n*(1/(d*(d + e*x^r) ) + Log[x^r]/d^2 - Log[d + e*x^r]/d^2))/(2*e*r^2)))/d) + (-((e*((x^r*(a + b*Log[c*x^n]))/(d*r*(d + e*x^r)) - (b*n*Log[d + e*x^r])/(d*e*r^2)))/d) + ( -(((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d*r)) + (b*n*PolyLog[2, -(d/(e* x^r))])/(d*r^2))/d)/d
3.5.26.3.1 Defintions of rubi rules used
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*L og[c*x^n])^p/(e*r*(q + 1))), x] - Simp[b*f^m*n*(p/(e*r*(q + 1))) Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d , e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || G tQ[f, 0]) && NeQ[r, n] && 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_)^(r_.))^ (q_))/(x_), x_Symbol] :> Simp[1/d Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x ^n])^p/x), x], x] - Simp[e/d Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n ])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1 ]
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 = 4.00 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.80
| method | result | size |
| risch | \(\frac {b \ln \left (d +e \,x^{r}\right ) n \ln \left (x \right )}{r \,d^{3}}-\frac {b \ln \left (d +e \,x^{r}\right ) \ln \left (x^{n}\right )}{r \,d^{3}}-\frac {b n \ln \left (x \right )}{r \,d^{2} \left (d +e \,x^{r}\right )}+\frac {b \ln \left (x^{n}\right )}{r \,d^{2} \left (d +e \,x^{r}\right )}-\frac {b n \ln \left (x \right )}{2 r d \left (d +e \,x^{r}\right )^{2}}+\frac {b \ln \left (x^{n}\right )}{2 r d \left (d +e \,x^{r}\right )^{2}}-\frac {b \ln \left (x^{r}\right ) n \ln \left (x \right )}{r \,d^{3}}+\frac {b \ln \left (x^{r}\right ) \ln \left (x^{n}\right )}{r \,d^{3}}+\frac {3 b n \ln \left (d +e \,x^{r}\right )}{2 d^{3} r^{2}}-\frac {b n e \ln \left (x \right ) x^{r}}{r \,d^{3} \left (d +e \,x^{r}\right )}-\frac {b n \operatorname {dilog}\left (\frac {d +e \,x^{r}}{d}\right )}{r^{2} d^{3}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {d +e \,x^{r}}{d}\right )}{r \,d^{3}}-\frac {b n}{2 d^{2} r^{2} \left (d +e \,x^{r}\right )}-\frac {b n \,e^{2} \ln \left (x \right ) x^{2 r}}{2 r \,d^{3} \left (d +e \,x^{r}\right )^{2}}-\frac {b n e \ln \left (x \right ) x^{r}}{r \,d^{2} \left (d +e \,x^{r}\right )^{2}}+\frac {b n \ln \left (x \right )^{2}}{2 d^{3}}+\frac {\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d +e \,x^{r}\right )}{d^{3}}+\frac {1}{d^{2} \left (d +e \,x^{r}\right )}+\frac {1}{2 d \left (d +e \,x^{r}\right )^{2}}+\frac {\ln \left (x^{r}\right )}{d^{3}}\right )}{r}\) | \(473\) |
b/r/d^3*ln(d+e*x^r)*n*ln(x)-b/r/d^3*ln(d+e*x^r)*ln(x^n)-b/r/d^2/(d+e*x^r)* n*ln(x)+b/r/d^2/(d+e*x^r)*ln(x^n)-1/2*b/r/d/(d+e*x^r)^2*n*ln(x)+1/2*b/r/d/ (d+e*x^r)^2*ln(x^n)-b/r/d^3*ln(x^r)*n*ln(x)+b/r/d^3*ln(x^r)*ln(x^n)+3/2*b* n*ln(d+e*x^r)/d^3/r^2-b/r*n*e/d^3*ln(x)*x^r/(d+e*x^r)-b/r^2*n/d^3*dilog((d +e*x^r)/d)-b/r*n/d^3*ln(x)*ln((d+e*x^r)/d)-1/2*b*n/d^2/r^2/(d+e*x^r)-1/2*b /r*n*e^2/d^3*ln(x)*(x^r)^2/(d+e*x^r)^2-b/r*n*e/d^2*ln(x)*x^r/(d+e*x^r)^2+1 /2*b*n/d^3*ln(x)^2+(-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)/r*(-1/d^3*ln(d+e*x^r)+1/d^2/(d+e*x^r)+1/ 2/d/(d+e*x^r)^2+1/d^3*ln(x^r))
Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (160) = 320\).
Time = 0.28 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.37 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\frac {b d^{2} n r^{2} \log \left (x\right )^{2} + 3 \, b d^{2} r \log \left (c\right ) - b d^{2} n + 3 \, a d^{2} r + {\left (b e^{2} n r^{2} \log \left (x\right )^{2} + {\left (2 \, b e^{2} r^{2} \log \left (c\right ) - 3 \, b e^{2} n r + 2 \, a e^{2} r^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + {\left (2 \, b d e n r^{2} \log \left (x\right )^{2} + 2 \, b d e r \log \left (c\right ) - b d e n + 2 \, a d e r + 4 \, {\left (b d e r^{2} \log \left (c\right ) - b d e n r + a d e r^{2}\right )} \log \left (x\right )\right )} x^{r} - 2 \, {\left (b e^{2} n x^{2 \, r} + 2 \, b d e n x^{r} + b d^{2} n\right )} {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - {\left (2 \, b d^{2} r \log \left (c\right ) - 3 \, b d^{2} n + 2 \, a d^{2} r + {\left (2 \, b e^{2} r \log \left (c\right ) - 3 \, b e^{2} n + 2 \, a e^{2} r\right )} x^{2 \, r} + 2 \, {\left (2 \, b d e r \log \left (c\right ) - 3 \, b d e n + 2 \, a d e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b d^{2} r^{2} \log \left (c\right ) + a d^{2} r^{2}\right )} \log \left (x\right ) - 2 \, {\left (b e^{2} n r x^{2 \, r} \log \left (x\right ) + 2 \, b d e n r x^{r} \log \left (x\right ) + b d^{2} n r \log \left (x\right )\right )} \log \left (\frac {e x^{r} + d}{d}\right )}{2 \, {\left (d^{3} e^{2} r^{2} x^{2 \, r} + 2 \, d^{4} e r^{2} x^{r} + d^{5} r^{2}\right )}} \]
1/2*(b*d^2*n*r^2*log(x)^2 + 3*b*d^2*r*log(c) - b*d^2*n + 3*a*d^2*r + (b*e^ 2*n*r^2*log(x)^2 + (2*b*e^2*r^2*log(c) - 3*b*e^2*n*r + 2*a*e^2*r^2)*log(x) )*x^(2*r) + (2*b*d*e*n*r^2*log(x)^2 + 2*b*d*e*r*log(c) - b*d*e*n + 2*a*d*e *r + 4*(b*d*e*r^2*log(c) - b*d*e*n*r + a*d*e*r^2)*log(x))*x^r - 2*(b*e^2*n *x^(2*r) + 2*b*d*e*n*x^r + b*d^2*n)*dilog(-(e*x^r + d)/d + 1) - (2*b*d^2*r *log(c) - 3*b*d^2*n + 2*a*d^2*r + (2*b*e^2*r*log(c) - 3*b*e^2*n + 2*a*e^2* r)*x^(2*r) + 2*(2*b*d*e*r*log(c) - 3*b*d*e*n + 2*a*d*e*r)*x^r)*log(e*x^r + d) + 2*(b*d^2*r^2*log(c) + a*d^2*r^2)*log(x) - 2*(b*e^2*n*r*x^(2*r)*log(x ) + 2*b*d*e*n*r*x^r*log(x) + b*d^2*n*r*log(x))*log((e*x^r + d)/d))/(d^3*e^ 2*r^2*x^(2*r) + 2*d^4*e*r^2*x^r + d^5*r^2)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{3} x} \,d x } \]
1/2*a*((2*e*x^r + 3*d)/(d^2*e^2*r*x^(2*r) + 2*d^3*e*r*x^r + d^4*r) + 2*log (x)/d^3 - 2*log((e*x^r + d)/e)/(d^3*r)) + b*integrate((log(c) + log(x^n))/ (e^3*x*x^(3*r) + 3*d*e^2*x*x^(2*r) + 3*d^2*e*x*x^r + d^3*x), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x^r\right )}^3} \,d x \]