Integrand size = 26, antiderivative size = 161 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )+6 b^3 m n^3 \operatorname {PolyLog}\left (5,-\frac {f x}{e}\right ) \]
1/4*(a+b*ln(c*x^n))^4*ln(d*(f*x+e)^m)/b/n-1/4*m*(a+b*ln(c*x^n))^4*ln(1+f*x /e)/b/n-m*(a+b*ln(c*x^n))^3*polylog(2,-f*x/e)+3*b*m*n*(a+b*ln(c*x^n))^2*po lylog(3,-f*x/e)-6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(4,-f*x/e)+6*b^3*m*n^3* polylog(5,-f*x/e)
Leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(161)=322\).
Time = 0.15 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.74 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=a^3 \log (x) \log \left (d (e+f x)^m\right )-\frac {3}{2} a^2 b n \log ^2(x) \log \left (d (e+f x)^m\right )+a b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )-\frac {1}{4} b^3 n^3 \log ^4(x) \log \left (d (e+f x)^m\right )+3 a^2 b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac {3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 \log (x) \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-a^3 m \log (x) \log \left (1+\frac {f x}{e}\right )+\frac {3}{2} a^2 b m n \log ^2(x) \log \left (1+\frac {f x}{e}\right )-a b^2 m n^2 \log ^3(x) \log \left (1+\frac {f x}{e}\right )+\frac {1}{4} b^3 m n^3 \log ^4(x) \log \left (1+\frac {f x}{e}\right )-3 a^2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+3 a b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^3 m n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-3 a b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+\frac {3}{2} b^3 m n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^3 m \log (x) \log ^3\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-m \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )-6 a b^2 m n^2 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )-6 b^3 m n^2 \log \left (c x^n\right ) \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )+6 b^3 m n^3 \operatorname {PolyLog}\left (5,-\frac {f x}{e}\right ) \]
a^3*Log[x]*Log[d*(e + f*x)^m] - (3*a^2*b*n*Log[x]^2*Log[d*(e + f*x)^m])/2 + a*b^2*n^2*Log[x]^3*Log[d*(e + f*x)^m] - (b^3*n^3*Log[x]^4*Log[d*(e + f*x )^m])/4 + 3*a^2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x)^m] - 3*a*b^2*n*Log[x]^ 2*Log[c*x^n]*Log[d*(e + f*x)^m] + b^3*n^2*Log[x]^3*Log[c*x^n]*Log[d*(e + f *x)^m] + 3*a*b^2*Log[x]*Log[c*x^n]^2*Log[d*(e + f*x)^m] - (3*b^3*n*Log[x]^ 2*Log[c*x^n]^2*Log[d*(e + f*x)^m])/2 + b^3*Log[x]*Log[c*x^n]^3*Log[d*(e + f*x)^m] - a^3*m*Log[x]*Log[1 + (f*x)/e] + (3*a^2*b*m*n*Log[x]^2*Log[1 + (f *x)/e])/2 - a*b^2*m*n^2*Log[x]^3*Log[1 + (f*x)/e] + (b^3*m*n^3*Log[x]^4*Lo g[1 + (f*x)/e])/4 - 3*a^2*b*m*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 3*a*b^2 *m*n*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/e] - b^3*m*n^2*Log[x]^3*Log[c*x^n]* Log[1 + (f*x)/e] - 3*a*b^2*m*Log[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] + (3*b^3 *m*n*Log[x]^2*Log[c*x^n]^2*Log[1 + (f*x)/e])/2 - b^3*m*Log[x]*Log[c*x^n]^3 *Log[1 + (f*x)/e] - m*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x)/e)] + 3*b*m* n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x)/e)] - 6*a*b^2*m*n^2*PolyLog[4, - ((f*x)/e)] - 6*b^3*m*n^2*Log[c*x^n]*PolyLog[4, -((f*x)/e)] + 6*b^3*m*n^3*P olyLog[5, -((f*x)/e)]
Time = 0.66 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2822, 2754, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2822 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \int \frac {\left (a+b \log \left (c x^n\right )\right )^4}{e+f x}dx}{4 b n}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {f x}{e}+1\right )}{x}dx}{f}\right )}{4 b n}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{x}dx\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}\left (4,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{x}dx\right )\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}\left (4,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \operatorname {PolyLog}\left (5,-\frac {f x}{e}\right )\right )\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\) |
((a + b*Log[c*x^n])^4*Log[d*(e + f*x)^m])/(4*b*n) - (f*m*(((a + b*Log[c*x^ n])^4*Log[1 + (f*x)/e])/f - (4*b*n*(-((a + b*Log[c*x^n])^3*PolyLog[2, -((f *x)/e)]) + 3*b*n*((a + b*Log[c*x^n])^2*PolyLog[3, -((f*x)/e)] - 2*b*n*((a + b*Log[c*x^n])*PolyLog[4, -((f*x)/e)] - b*n*PolyLog[5, -((f*x)/e)]))))/f) )/(4*b*n)
3.1.87.3.1 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[(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 150.14 (sec) , antiderivative size = 15171, normalized size of antiderivative = 94.23
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x} \,d x } \]
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)*log((f*x + e)^m*d)/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x} \,d x } \]
-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 + e)^m) - integrate(-1/4*(b^3*f *m*n^3*x*log(x)^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*f*m*n^2*log(c) + a*b^2 *f*m*n^2)*x*log(x)^3 + 6*(b^3*f*m*n*log(c)^2 + 2*a*b^2*f*m*n*log(c) + a^2* b*f*m*n)*x*log(x)^2 - 4*(b^3*f*m*x*log(x) - b^3*f*x*log(d) - b^3*e*log(d)) *log(x^n)^3 - 4*(b^3*f*m*log(c)^3 + 3*a*b^2*f*m*log(c)^2 + 3*a^2*b*f*m*log (c) + a^3*f*m)*x*log(x) + 6*(b^3*f*m*n*x*log(x)^2 + 2*b^3*e*log(c)*log(d) + 2*a*b^2*e*log(d) - 2*(b^3*f*m*log(c) + a*b^2*f*m)*x*log(x) + 2*(b^3*f*lo g(c)*log(d) + a*b^2*f*log(d))*x)*log(x^n)^2 + 4*(b^3*f*log(c)^3*log(d) + 3 *a*b^2*f*log(c)^2*log(d) + 3*a^2*b*f*log(c)*log(d) + a^3*f*log(d))*x - 4*( b^3*f*m*n^2*x*log(x)^3 - 3*b^3*e*log(c)^2*log(d) - 6*a*b^2*e*log(c)*log(d) - 3*a^2*b*e*log(d) - 3*(b^3*f*m*n*log(c) + a*b^2*f*m*n)*x*log(x)^2 + 3*(b ^3*f*m*log(c)^2 + 2*a*b^2*f*m*log(c) + a^2*b*f*m)*x*log(x) - 3*(b^3*f*log( c)^2*log(d) + 2*a*b^2*f*log(c)*log(d) + a^2*b*f*log(d))*x)*log(x^n))/(f*x^ 2 + e*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]