Integrand size = 26, antiderivative size = 131 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )-2 b^2 m n^2 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right ) \]
1/3*(a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)/b/n-1/3*m*(a+b*ln(c*x^n))^3*ln(1+f*x /e)/b/n-m*(a+b*ln(c*x^n))^2*polylog(2,-f*x/e)+2*b*m*n*(a+b*ln(c*x^n))*poly log(3,-f*x/e)-2*b^2*m*n^2*polylog(4,-f*x/e)
Leaf count is larger than twice the leaf count of optimal. \(329\) vs. \(2(131)=262\).
Time = 0.11 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.51 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=a^2 \log (x) \log \left (d (e+f x)^m\right )-a b n \log ^2(x) \log \left (d (e+f x)^m\right )+\frac {1}{3} b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )+2 a b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-a^2 m \log (x) \log \left (1+\frac {f x}{e}\right )+a b m n \log ^2(x) \log \left (1+\frac {f x}{e}\right )-\frac {1}{3} b^2 m n^2 \log ^3(x) \log \left (1+\frac {f x}{e}\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-m \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )-2 b^2 m n^2 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right ) \]
a^2*Log[x]*Log[d*(e + f*x)^m] - a*b*n*Log[x]^2*Log[d*(e + f*x)^m] + (b^2*n ^2*Log[x]^3*Log[d*(e + f*x)^m])/3 + 2*a*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x )^m] - b^2*n*Log[x]^2*Log[c*x^n]*Log[d*(e + f*x)^m] + b^2*Log[x]*Log[c*x^n ]^2*Log[d*(e + f*x)^m] - a^2*m*Log[x]*Log[1 + (f*x)/e] + a*b*m*n*Log[x]^2* Log[1 + (f*x)/e] - (b^2*m*n^2*Log[x]^3*Log[1 + (f*x)/e])/3 - 2*a*b*m*Log[x ]*Log[c*x^n]*Log[1 + (f*x)/e] + b^2*m*n*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/ e] - b^2*m*Log[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] - m*(a + b*Log[c*x^n])^2*P olyLog[2, -((f*x)/e)] + 2*b*m*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)] - 2*b^2*m*n^2*PolyLog[4, -((f*x)/e)]
Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2822, 2754, 2821, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2822 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {f m \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{e+f x}dx}{3 b n}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {f x}{e}+1\right )}{x}dx}{f}\right )}{3 b n}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {3 b n \left (2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2\right )}{f}\right )}{3 b n}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {3 b n \left (2 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{x}dx\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2\right )}{f}\right )}{3 b n}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {3 b n \left (2 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2\right )}{f}\right )}{3 b n}\) |
((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/(3*b*n) - (f*m*(((a + b*Log[c*x^ n])^3*Log[1 + (f*x)/e])/f - (3*b*n*(-((a + b*Log[c*x^n])^2*PolyLog[2, -((f *x)/e)]) + 2*b*n*((a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)] - b*n*PolyLog[ 4, -((f*x)/e)])))/f))/(3*b*n)
3.1.81.3.1 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 = 33.19 (sec) , antiderivative size = 3957, normalized size of antiderivative = 30.21
-m*dilog((f*x+e)/e)*ln(c)^2*b^2-m*dilog((f*x+e)/e)*ln(x^n)^2*b^2-m*ln(x)*l n((f*x+e)/e)*a^2-m*dilog((f*x+e)/e)*a^2-I*m*dilog((f*x+e)/e)*Pi*a*b*csgn(I *c)*csgn(I*c*x^n)^2-I*m*dilog((f*x+e)/e)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^ 2-2*m*b^2*n*ln(x)*ln(x^n)*polylog(2,-f*x/e)-2*m*b*n*ln(x)*polylog(2,-f*x/e )*a+1/4*m*dilog((f*x+e)/e)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-1/2*m*dilo g((f*x+e)/e)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/4*m*dilog((f*x+e)/e)*Pi^ 2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-1/2*m*dilog((f*x+e)/e)*Pi^2*b^2*csgn(I *x^n)*csgn(I*c*x^n)^5+2*m*dilog((f*x+e)/e)*ln(x)*ln(c)*b^2*n+2*m*dilog((f* x+e)/e)*ln(x)*ln(x^n)*b^2*n+2*m*dilog((f*x+e)/e)*ln(x)*a*b*n+1/4*m*ln(x)*l n((f*x+e)/e)*Pi^2*b^2*csgn(I*c*x^n)^6+2*m*ln(x)^2*ln((f*x+e)/e)*ln(c)*b^2* n+2*m*ln(x)^2*ln((f*x+e)/e)*ln(x^n)*b^2*n+2*m*ln(x)^2*ln((f*x+e)/e)*a*b*n- 2*m*ln(x)*ln((f*x+e)/e)*ln(c)*ln(x^n)*b^2-2*m*ln(x)*ln((f*x+e)/e)*ln(c)*a* b-2*m*ln(x)*ln((f*x+e)/e)*ln(x^n)*a*b-m*b^2*n*ln(x)^2*ln(c)*ln(1+f*x/e)-m* b^2*n*ln(x)^2*ln(x^n)*ln(1+f*x/e)-m*b*n*ln(x)^2*ln(1+f*x/e)*a-2*m*b^2*n*ln (x)*ln(c)*polylog(2,-f*x/e)+(ln(c)^2*ln(x)*b^2+1/3*b^2*n^2*ln(x)^3+(-I*Pi* b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(x)+I*Pi*b^2*csgn(I*c)*csgn(I*c* x^n)^2*ln(x)+I*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(x)-I*Pi*b^2*csgn(I*c* x^n)^3*ln(x)-b^2*n*ln(x)^2+2*ln(c)*b^2*ln(x)+2*a*b*ln(x))*ln(x^n)-1/4*Pi^2 *ln(x)*b^2*csgn(I*c*x^n)^6+ln(x)*a^2+b^2*ln(x)*ln(x^n)^2-I*ln(c)*Pi*ln(x)* b^2*csgn(I*c*x^n)^3-I*Pi*ln(x)*a*b*csgn(I*c*x^n)^3+1/2*I*ln(x)^2*Pi*b^2...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x} \,d x } \]
1/3*(b^2*n^2*log(x)^3 + 3*b^2*log(x)*log(x^n)^2 - 3*(b^2*n*log(c) + a*b*n) *log(x)^2 - 3*(b^2*n*log(x)^2 - 2*(b^2*log(c) + a*b)*log(x))*log(x^n) + 3* (b^2*log(c)^2 + 2*a*b*log(c) + a^2)*log(x))*log((f*x + e)^m) - integrate(1 /3*(b^2*f*m*n^2*x*log(x)^3 - 3*b^2*e*log(c)^2*log(d) - 6*a*b*e*log(c)*log( d) - 3*a^2*e*log(d) - 3*(b^2*f*m*n*log(c) + a*b*f*m*n)*x*log(x)^2 + 3*(b^2 *f*m*log(c)^2 + 2*a*b*f*m*log(c) + a^2*f*m)*x*log(x) + 3*(b^2*f*m*x*log(x) - b^2*f*x*log(d) - b^2*e*log(d))*log(x^n)^2 - 3*(b^2*f*log(c)^2*log(d) + 2*a*b*f*log(c)*log(d) + a^2*f*log(d))*x - 3*(b^2*f*m*n*x*log(x)^2 + 2*b^2* e*log(c)*log(d) + 2*a*b*e*log(d) - 2*(b^2*f*m*log(c) + a*b*f*m)*x*log(x) + 2*(b^2*f*log(c)*log(d) + a*b*f*log(d))*x)*log(x^n))/(f*x^2 + e*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]