Integrand size = 26, antiderivative size = 823 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\frac {1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )+\frac {1}{12} b^2 m n^2 \left (\log ^4\left (-\frac {e x}{d}\right )+6 \log ^2\left (-\frac {e x}{d}\right ) \log ^2\left (-\frac {e x}{d+e x}\right )-4 \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {d}{d+e x}\right )\right ) \log ^3\left (-\frac {e x}{d+e x}\right )+\log ^4\left (-\frac {e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+4 \left (2 \log ^3\left (-\frac {e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (1+\frac {e x}{d}\right )-4 \log ^2\left (-\frac {e x}{d}\right ) \log \left (-\frac {e x}{d+e x}\right ) \left (\log \left (-\frac {e x}{d}\right )+3 \log \left (1+\frac {e x}{d}\right )\right )+12 \left (\log ^2\left (-\frac {e x}{d}\right )-2 \log \left (-\frac {e x}{d}\right ) \left (\log \left (-\frac {e x}{d+e x}\right )+\log \left (1+\frac {e x}{d}\right )\right )+2 \log (x) \left (-\log (d+e x)+\log \left (1+\frac {e x}{d}\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 \log ^2\left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (2,\frac {e x}{d+e x}\right )+12 \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {e x}{d+e x}\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log \left (1+\frac {e x}{d}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log \left (-\frac {e x}{d+e x}\right )+\log (d+e x)\right ) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+24 \log \left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (3,\frac {e x}{d+e x}\right )+24 \left (-\log (x)+\log \left (-\frac {e x}{d+e x}\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )-24 \left (\operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (4,\frac {e x}{d+e x}\right )-\operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )\right )\right ) \]
1/2*m*ln(x)^2*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2+ln(x)*(-m*ln(x)+ln(f*x ^m))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2+2*b*n*(-m*ln(x)+ln(f*x^m))*(a-b *n*ln(e*x+d)+b*ln(c*(e*x+d)^n))*(ln(x)*(ln(e*x+d)-ln(1+e*x/d))-polylog(2,- e*x/d))+2*b*m*n*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))*(1/2*ln(x)^2*(ln(e*x+d )-ln(1+e*x/d))-ln(x)*polylog(2,-e*x/d)+polylog(3,-e*x/d))-b^2*n^2*(m*ln(x) -ln(f*x^m))*(ln(-e*x/d)*ln(e*x+d)^2+2*ln(e*x+d)*polylog(2,1+e*x/d)-2*polyl og(3,1+e*x/d))+1/12*b^2*m*n^2*(ln(-e*x/d)^4+6*ln(-e*x/d)^2*ln(-e*x/(e*x+d) )^2-4*(ln(-e*x/d)+ln(d/(e*x+d)))*ln(-e*x/(e*x+d))^3+ln(-e*x/(e*x+d))^4+6*l n(x)^2*ln(e*x+d)^2+4*(2*ln(-e*x/d)^3-3*ln(x)^2*ln(e*x+d))*ln(1+e*x/d)+6*(l n(x)-ln(-e*x/d))*(ln(x)+3*ln(-e*x/d))*ln(1+e*x/d)^2-4*ln(-e*x/d)^2*ln(-e*x /(e*x+d))*(ln(-e*x/d)+3*ln(1+e*x/d))+12*(ln(-e*x/d)^2-2*ln(-e*x/d)*(ln(-e* x/(e*x+d))+ln(1+e*x/d))+2*ln(x)*(-ln(e*x+d)+ln(1+e*x/d)))*polylog(2,-e*x/d )-12*ln(-e*x/(e*x+d))^2*polylog(2,e*x/(e*x+d))+12*(ln(-e*x/d)-ln(-e*x/(e*x +d)))^2*polylog(2,1+e*x/d)+24*(ln(x)-ln(-e*x/d))*ln(1+e*x/d)*polylog(2,1+e *x/d)+24*(ln(-e*x/(e*x+d))+ln(e*x+d))*polylog(3,-e*x/d)+24*ln(-e*x/(e*x+d) )*polylog(3,e*x/(e*x+d))+24*(-ln(x)+ln(-e*x/(e*x+d)))*polylog(3,1+e*x/d)-2 4*polylog(4,-e*x/d)-24*polylog(4,e*x/(e*x+d))+24*polylog(4,1+e*x/d))
Time = 0.23 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\frac {1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )+\frac {1}{12} b^2 m n^2 \left (\log ^4\left (-\frac {e x}{d}\right )+6 \log ^2\left (-\frac {e x}{d}\right ) \log ^2\left (-\frac {e x}{d+e x}\right )-4 \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {d}{d+e x}\right )\right ) \log ^3\left (-\frac {e x}{d+e x}\right )+\log ^4\left (-\frac {e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+4 \left (2 \log ^3\left (-\frac {e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (1+\frac {e x}{d}\right )-4 \log ^2\left (-\frac {e x}{d}\right ) \log \left (-\frac {e x}{d+e x}\right ) \left (\log \left (-\frac {e x}{d}\right )+3 \log \left (1+\frac {e x}{d}\right )\right )+12 \left (\log ^2\left (-\frac {e x}{d}\right )-2 \log \left (-\frac {e x}{d}\right ) \left (\log \left (-\frac {e x}{d+e x}\right )+\log \left (1+\frac {e x}{d}\right )\right )+2 \log (x) \left (-\log (d+e x)+\log \left (1+\frac {e x}{d}\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 \log ^2\left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (2,\frac {e x}{d+e x}\right )+12 \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {e x}{d+e x}\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log \left (1+\frac {e x}{d}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+24 \left (\log \left (-\frac {e x}{d+e x}\right )+\log (d+e x)\right ) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+24 \log \left (-\frac {e x}{d+e x}\right ) \operatorname {PolyLog}\left (3,\frac {e x}{d+e x}\right )+24 \left (-\log (x)+\log \left (-\frac {e x}{d+e x}\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )-24 \left (\operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )+\operatorname {PolyLog}\left (4,\frac {e x}{d+e x}\right )-\operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )\right )\right ) \]
(m*Log[x]^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/2 + Log[x]*(- (m*Log[x]) + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 2*b*n*(-(m*Log[x]) + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x )^n])*(Log[x]*(Log[d + e*x] - Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) + 2*b*m*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((Log[x]^2*(Log[d + e*x] - Log[1 + (e*x)/d]))/2 - Log[x]*PolyLog[2, -((e*x)/d)] + PolyLog[3, -((e*x)/d)]) - b^2*n^2*(m*Log[x] - Log[f*x^m])*(Log[-((e*x)/d)]*Log[d + e *x]^2 + 2*Log[d + e*x]*PolyLog[2, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d] ) + (b^2*m*n^2*(Log[-((e*x)/d)]^4 + 6*Log[-((e*x)/d)]^2*Log[-((e*x)/(d + e *x))]^2 - 4*(Log[-((e*x)/d)] + Log[d/(d + e*x)])*Log[-((e*x)/(d + e*x))]^3 + Log[-((e*x)/(d + e*x))]^4 + 6*Log[x]^2*Log[d + e*x]^2 + 4*(2*Log[-((e*x )/d)]^3 - 3*Log[x]^2*Log[d + e*x])*Log[1 + (e*x)/d] + 6*(Log[x] - Log[-((e *x)/d)])*(Log[x] + 3*Log[-((e*x)/d)])*Log[1 + (e*x)/d]^2 - 4*Log[-((e*x)/d )]^2*Log[-((e*x)/(d + e*x))]*(Log[-((e*x)/d)] + 3*Log[1 + (e*x)/d]) + 12*( Log[-((e*x)/d)]^2 - 2*Log[-((e*x)/d)]*(Log[-((e*x)/(d + e*x))] + Log[1 + ( e*x)/d]) + 2*Log[x]*(-Log[d + e*x] + Log[1 + (e*x)/d]))*PolyLog[2, -((e*x) /d)] - 12*Log[-((e*x)/(d + e*x))]^2*PolyLog[2, (e*x)/(d + e*x)] + 12*(Log[ -((e*x)/d)] - Log[-((e*x)/(d + e*x))])^2*PolyLog[2, 1 + (e*x)/d] + 24*(Log [x] - Log[-((e*x)/d)])*Log[1 + (e*x)/d]*PolyLog[2, 1 + (e*x)/d] + 24*(Log[ -((e*x)/(d + e*x))] + Log[d + e*x])*PolyLog[3, -((e*x)/d)] + 24*Log[-((...
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.
| \(\displaystyle \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 2874 |
| \(\displaystyle \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 m}-\frac {b e n \int \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}dx}{m}\) |
| \(\Big \downarrow \) 2891 |
| \(\displaystyle \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 m}-\frac {b e n \int \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}dx}{m}\) |
3.4.70.3.1 Defintions of rubi rules used
Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[Log[f*x^m]^2*((a + b*Log[c*(d + e*x)^n] )^p/(2*m)), x] - Simp[b*e*n*(p/(2*m)) Int[Log[f*x^m]^2*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))^(q_.)*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^p*(f + g* Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r}, x]
\[\int \frac {\ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x}d x\]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x} \,d x } \]
integral((b^2*log((e*x + d)^n*c)^2*log(f*x^m) + 2*a*b*log((e*x + d)^n*c)*l og(f*x^m) + a^2*log(f*x^m))/x, x)
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x} \,d x } \]
-1/2*(b^2*m*log(x)^2 - 2*b^2*log(f)*log(x) - 2*b^2*log(x)*log(x^m))*log((e *x + d)^n)^2 - integrate(-(b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f) + (b^2*e*log(c)^2*log(f) + 2*a*b*e*log(c)*log(f) + a^2*e*log (f))*x + (b^2*e*m*n*x*log(x)^2 - 2*b^2*e*n*x*log(f)*log(x) + 2*b^2*d*log(c )*log(f) + 2*a*b*d*log(f) + 2*(b^2*e*log(c)*log(f) + a*b*e*log(f))*x - 2*( b^2*e*n*x*log(x) - b^2*d*log(c) - a*b*d - (b^2*e*log(c) + a*b*e)*x)*log(x^ m))*log((e*x + d)^n) + (b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d + (b^2*e*l og(c)^2 + 2*a*b*e*log(c) + a^2*e)*x)*log(x^m))/(e*x^2 + d*x), x)
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx=\int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x} \,d x \]