Integrand size = 28, antiderivative size = 185 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x^m}{e}\right )}{4 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )}{m}+\frac {3 b n r \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{m^2}-\frac {6 b^2 n^2 r \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-\frac {f x^m}{e}\right )}{m^3}+\frac {6 b^3 n^3 r \operatorname {PolyLog}\left (5,-\frac {f x^m}{e}\right )}{m^4} \]
1/4*(a+b*ln(c*x^n))^4*ln(d*(e+f*x^m)^r)/b/n-1/4*r*(a+b*ln(c*x^n))^4*ln(1+f *x^m/e)/b/n-r*(a+b*ln(c*x^n))^3*polylog(2,-f*x^m/e)/m+3*b*n*r*(a+b*ln(c*x^ n))^2*polylog(3,-f*x^m/e)/m^2-6*b^2*n^2*r*(a+b*ln(c*x^n))*polylog(4,-f*x^m /e)/m^3+6*b^3*n^3*r*polylog(5,-f*x^m/e)/m^4
Leaf count is larger than twice the leaf count of optimal. \(1395\) vs. \(2(185)=370\).
Time = 0.38 (sec) , antiderivative size = 1395, normalized size of antiderivative = 7.54 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx =\text {Too large to display} \]
-1/2*(a^2*b*m*n*r*Log[x]^3) + (3*a*b^2*m*n^2*r*Log[x]^4)/4 - (3*b^3*m*n^3* r*Log[x]^5)/10 - a*b^2*m*n*r*Log[x]^3*Log[c*x^n] + (3*b^3*m*n^2*r*Log[x]^4 *Log[c*x^n])/4 - (b^3*m*n*r*Log[x]^3*Log[c*x^n]^2)/2 - (3*a^2*b*n*r*Log[x] ^2*Log[1 + e/(f*x^m)])/2 + 2*a*b^2*n^2*r*Log[x]^3*Log[1 + e/(f*x^m)] - (3* b^3*n^3*r*Log[x]^4*Log[1 + e/(f*x^m)])/4 - 3*a*b^2*n*r*Log[x]^2*Log[c*x^n] *Log[1 + e/(f*x^m)] + 2*b^3*n^2*r*Log[x]^3*Log[c*x^n]*Log[1 + e/(f*x^m)] - (3*b^3*n*r*Log[x]^2*Log[c*x^n]^2*Log[1 + e/(f*x^m)])/2 - a^3*r*Log[x]*Log [e + f*x^m] + 3*a^2*b*n*r*Log[x]^2*Log[e + f*x^m] - 3*a*b^2*n^2*r*Log[x]^3 *Log[e + f*x^m] + b^3*n^3*r*Log[x]^4*Log[e + f*x^m] + (a^3*r*Log[-((f*x^m) /e)]*Log[e + f*x^m])/m - (3*a^2*b*n*r*Log[x]*Log[-((f*x^m)/e)]*Log[e + f*x ^m])/m + (3*a*b^2*n^2*r*Log[x]^2*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - (b^ 3*n^3*r*Log[x]^3*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - 3*a^2*b*r*Log[x]*Lo g[c*x^n]*Log[e + f*x^m] + 6*a*b^2*n*r*Log[x]^2*Log[c*x^n]*Log[e + f*x^m] - 3*b^3*n^2*r*Log[x]^3*Log[c*x^n]*Log[e + f*x^m] + (3*a^2*b*r*Log[-((f*x^m) /e)]*Log[c*x^n]*Log[e + f*x^m])/m - (6*a*b^2*n*r*Log[x]*Log[-((f*x^m)/e)]* Log[c*x^n]*Log[e + f*x^m])/m + (3*b^3*n^2*r*Log[x]^2*Log[-((f*x^m)/e)]*Log [c*x^n]*Log[e + f*x^m])/m - 3*a*b^2*r*Log[x]*Log[c*x^n]^2*Log[e + f*x^m] + 3*b^3*n*r*Log[x]^2*Log[c*x^n]^2*Log[e + f*x^m] + (3*a*b^2*r*Log[-((f*x^m) /e)]*Log[c*x^n]^2*Log[e + f*x^m])/m - (3*b^3*n*r*Log[x]*Log[-((f*x^m)/e)]* Log[c*x^n]^2*Log[e + f*x^m])/m - b^3*r*Log[x]*Log[c*x^n]^3*Log[e + f*x^...
Time = 0.80 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2822, 2775, 2821, 2830, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2822 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {f m r \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )^4}{f x^m+e}dx}{4 b n}\) |
| \(\Big \downarrow \) 2775 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {f m r \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f m}-\frac {4 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {f x^m}{e}+1\right )}{x}dx}{f m}\right )}{4 b n}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {f m r \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f m}-\frac {4 b n \left (\frac {3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )}{x}dx}{m}-\frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\right )}{f m}\right )}{4 b n}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {f m r \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f m}-\frac {4 b n \left (\frac {3 b n \left (\frac {\operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{x}dx}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\right )}{f m}\right )}{4 b n}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {f m r \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f m}-\frac {4 b n \left (\frac {3 b n \left (\frac {\operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (4,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \int \frac {\operatorname {PolyLog}\left (4,-\frac {f x^m}{e}\right )}{x}dx}{m}\right )}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\right )}{f m}\right )}{4 b n}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^m\right )^r\right )}{4 b n}-\frac {f m r \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f m}-\frac {4 b n \left (\frac {3 b n \left (\frac {\operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (4,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \operatorname {PolyLog}\left (5,-\frac {f x^m}{e}\right )}{m^2}\right )}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\right )}{f m}\right )}{4 b n}\) |
((a + b*Log[c*x^n])^4*Log[d*(e + f*x^m)^r])/(4*b*n) - (f*m*r*(((a + b*Log[ c*x^n])^4*Log[1 + (f*x^m)/e])/(f*m) - (4*b*n*(-(((a + b*Log[c*x^n])^3*Poly Log[2, -((f*x^m)/e)])/m) + (3*b*n*(((a + b*Log[c*x^n])^2*PolyLog[3, -((f*x ^m)/e)])/m - (2*b*n*(((a + b*Log[c*x^n])*PolyLog[4, -((f*x^m)/e)])/m - (b* n*PolyLog[5, -((f*x^m)/e)])/m^2))/m))/m))/(f*m)))/(4*b*n)
3.2.39.3.1 Defintions of rubi rules used
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Simp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c* x^n])^p/(e*r)), x] - Simp[b*f^m*n*(p/(e*r)) Int[Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] & & EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
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[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_ .)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[ c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[f*m*(r/(b*n*(p + 1))) Int[x^(m - 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[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]
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (e +f \,x^{m}\right )^{r}\right )}{x}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (180) = 360\).
Time = 0.29 (sec) , antiderivative size = 765, normalized size of antiderivative = 4.14 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\frac {b^{3} m^{4} n^{3} \log \left (d\right ) \log \left (x\right )^{4} + 24 \, b^{3} n^{3} r {\rm polylog}\left (5, -\frac {f x^{m}}{e}\right ) + 4 \, {\left (b^{3} m^{4} n^{2} \log \left (c\right ) + a b^{2} m^{4} n^{2}\right )} \log \left (d\right ) \log \left (x\right )^{3} + 6 \, {\left (b^{3} m^{4} n \log \left (c\right )^{2} + 2 \, a b^{2} m^{4} n \log \left (c\right ) + a^{2} b m^{4} n\right )} \log \left (d\right ) \log \left (x\right )^{2} + 4 \, {\left (b^{3} m^{4} \log \left (c\right )^{3} + 3 \, a b^{2} m^{4} \log \left (c\right )^{2} + 3 \, a^{2} b m^{4} \log \left (c\right ) + a^{3} m^{4}\right )} \log \left (d\right ) \log \left (x\right ) - 4 \, {\left (b^{3} m^{3} n^{3} r \log \left (x\right )^{3} + b^{3} m^{3} r \log \left (c\right )^{3} + 3 \, a b^{2} m^{3} r \log \left (c\right )^{2} + 3 \, a^{2} b m^{3} r \log \left (c\right ) + a^{3} m^{3} r + 3 \, {\left (b^{3} m^{3} n^{2} r \log \left (c\right ) + a b^{2} m^{3} n^{2} r\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{3} m^{3} n r \log \left (c\right )^{2} + 2 \, a b^{2} m^{3} n r \log \left (c\right ) + a^{2} b m^{3} n r\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) + {\left (b^{3} m^{4} n^{3} r \log \left (x\right )^{4} + 4 \, {\left (b^{3} m^{4} n^{2} r \log \left (c\right ) + a b^{2} m^{4} n^{2} r\right )} \log \left (x\right )^{3} + 6 \, {\left (b^{3} m^{4} n r \log \left (c\right )^{2} + 2 \, a b^{2} m^{4} n r \log \left (c\right ) + a^{2} b m^{4} n r\right )} \log \left (x\right )^{2} + 4 \, {\left (b^{3} m^{4} r \log \left (c\right )^{3} + 3 \, a b^{2} m^{4} r \log \left (c\right )^{2} + 3 \, a^{2} b m^{4} r \log \left (c\right ) + a^{3} m^{4} r\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) - {\left (b^{3} m^{4} n^{3} r \log \left (x\right )^{4} + 4 \, {\left (b^{3} m^{4} n^{2} r \log \left (c\right ) + a b^{2} m^{4} n^{2} r\right )} \log \left (x\right )^{3} + 6 \, {\left (b^{3} m^{4} n r \log \left (c\right )^{2} + 2 \, a b^{2} m^{4} n r \log \left (c\right ) + a^{2} b m^{4} n r\right )} \log \left (x\right )^{2} + 4 \, {\left (b^{3} m^{4} r \log \left (c\right )^{3} + 3 \, a b^{2} m^{4} r \log \left (c\right )^{2} + 3 \, a^{2} b m^{4} r \log \left (c\right ) + a^{3} m^{4} r\right )} \log \left (x\right )\right )} \log \left (\frac {f x^{m} + e}{e}\right ) - 24 \, {\left (b^{3} m n^{3} r \log \left (x\right ) + b^{3} m n^{2} r \log \left (c\right ) + a b^{2} m n^{2} r\right )} {\rm polylog}\left (4, -\frac {f x^{m}}{e}\right ) + 12 \, {\left (b^{3} m^{2} n^{3} r \log \left (x\right )^{2} + b^{3} m^{2} n r \log \left (c\right )^{2} + 2 \, a b^{2} m^{2} n r \log \left (c\right ) + a^{2} b m^{2} n r + 2 \, {\left (b^{3} m^{2} n^{2} r \log \left (c\right ) + a b^{2} m^{2} n^{2} r\right )} \log \left (x\right )\right )} {\rm polylog}\left (3, -\frac {f x^{m}}{e}\right )}{4 \, m^{4}} \]
1/4*(b^3*m^4*n^3*log(d)*log(x)^4 + 24*b^3*n^3*r*polylog(5, -f*x^m/e) + 4*( b^3*m^4*n^2*log(c) + a*b^2*m^4*n^2)*log(d)*log(x)^3 + 6*(b^3*m^4*n*log(c)^ 2 + 2*a*b^2*m^4*n*log(c) + a^2*b*m^4*n)*log(d)*log(x)^2 + 4*(b^3*m^4*log(c )^3 + 3*a*b^2*m^4*log(c)^2 + 3*a^2*b*m^4*log(c) + a^3*m^4)*log(d)*log(x) - 4*(b^3*m^3*n^3*r*log(x)^3 + b^3*m^3*r*log(c)^3 + 3*a*b^2*m^3*r*log(c)^2 + 3*a^2*b*m^3*r*log(c) + a^3*m^3*r + 3*(b^3*m^3*n^2*r*log(c) + a*b^2*m^3*n^ 2*r)*log(x)^2 + 3*(b^3*m^3*n*r*log(c)^2 + 2*a*b^2*m^3*n*r*log(c) + a^2*b*m ^3*n*r)*log(x))*dilog(-(f*x^m + e)/e + 1) + (b^3*m^4*n^3*r*log(x)^4 + 4*(b ^3*m^4*n^2*r*log(c) + a*b^2*m^4*n^2*r)*log(x)^3 + 6*(b^3*m^4*n*r*log(c)^2 + 2*a*b^2*m^4*n*r*log(c) + a^2*b*m^4*n*r)*log(x)^2 + 4*(b^3*m^4*r*log(c)^3 + 3*a*b^2*m^4*r*log(c)^2 + 3*a^2*b*m^4*r*log(c) + a^3*m^4*r)*log(x))*log( f*x^m + e) - (b^3*m^4*n^3*r*log(x)^4 + 4*(b^3*m^4*n^2*r*log(c) + a*b^2*m^4 *n^2*r)*log(x)^3 + 6*(b^3*m^4*n*r*log(c)^2 + 2*a*b^2*m^4*n*r*log(c) + a^2* b*m^4*n*r)*log(x)^2 + 4*(b^3*m^4*r*log(c)^3 + 3*a*b^2*m^4*r*log(c)^2 + 3*a ^2*b*m^4*r*log(c) + a^3*m^4*r)*log(x))*log((f*x^m + e)/e) - 24*(b^3*m*n^3* r*log(x) + b^3*m*n^2*r*log(c) + a*b^2*m*n^2*r)*polylog(4, -f*x^m/e) + 12*( b^3*m^2*n^3*r*log(x)^2 + b^3*m^2*n*r*log(c)^2 + 2*a*b^2*m^2*n*r*log(c) + a ^2*b*m^2*n*r + 2*(b^3*m^2*n^2*r*log(c) + a*b^2*m^2*n^2*r)*log(x))*polylog( 3, -f*x^m/e))/m^4
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{m} + e\right )}^{r} d\right )}{x} \,d x } \]
-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b ^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2 + 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^ 2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b^3 *n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2* a*b^2*log(c) + a^2*b)*log(x))*log(x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^ 2 + 3*a^2*b*log(c) + a^3)*log(x))*log((f*x^m + e)^r) - integrate(-1/4*(4*b ^3*e*log(c)^3*log(d) + 12*a*b^2*e*log(c)^2*log(d) + 12*a^2*b*e*log(c)*log( d) + 4*a^3*e*log(d) + 4*(b^3*e*log(d) - (b^3*f*m*r*log(x) - b^3*f*log(d))* x^m)*log(x^n)^3 + 6*(2*b^3*e*log(c)*log(d) + 2*a*b^2*e*log(d) + (b^3*f*m*n *r*log(x)^2 + 2*b^3*f*log(c)*log(d) + 2*a*b^2*f*log(d) - 2*(b^3*f*m*r*log( c) + a*b^2*f*m*r)*log(x))*x^m)*log(x^n)^2 + (b^3*f*m*n^3*r*log(x)^4 + 4*b^ 3*f*log(c)^3*log(d) + 12*a*b^2*f*log(c)^2*log(d) + 12*a^2*b*f*log(c)*log(d ) + 4*a^3*f*log(d) - 4*(b^3*f*m*n^2*r*log(c) + a*b^2*f*m*n^2*r)*log(x)^3 + 6*(b^3*f*m*n*r*log(c)^2 + 2*a*b^2*f*m*n*r*log(c) + a^2*b*f*m*n*r)*log(x)^ 2 - 4*(b^3*f*m*r*log(c)^3 + 3*a*b^2*f*m*r*log(c)^2 + 3*a^2*b*f*m*r*log(c) + a^3*f*m*r)*log(x))*x^m + 4*(3*b^3*e*log(c)^2*log(d) + 6*a*b^2*e*log(c)*l og(d) + 3*a^2*b*e*log(d) - (b^3*f*m*n^2*r*log(x)^3 - 3*b^3*f*log(c)^2*log( d) - 6*a*b^2*f*log(c)*log(d) - 3*a^2*b*f*log(d) - 3*(b^3*f*m*n*r*log(c) + a*b^2*f*m*n*r)*log(x)^2 + 3*(b^3*f*m*r*log(c)^2 + 2*a*b^2*f*m*r*log(c) + a ^2*b*f*m*r)*log(x))*x^m)*log(x^n))/(f*x*x^m + e*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{m} + e\right )}^{r} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^r\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]