Integrand size = 24, antiderivative size = 602 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=-\frac {a b d m n x}{2 e}+\frac {2 b^2 d m n^2 x}{e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {3 b^2 d^2 m n^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^2}-\frac {b^2 d^2 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e^2} \]
-1/2*a*b*d*m*n*x/e+2*b^2*d*m*n^2*x/e-2*b*d*m*n*(-b*n+a)*x/e-1/8*b^2*m*n^2* x^2-1/4*b^2*m*n^2*(e*x+d)^2/e^2-1/4*b^2*d^2*m*n^2*ln(x)/e^2+2*a*b*d*n*x*ln (f*x^m)/e-2*b^2*d*n^2*x*ln(f*x^m)/e+1/4*b^2*n^2*(e*x+d)^2*ln(f*x^m)/e^2-5/ 2*b^2*d*m*n*(e*x+d)*ln(c*(e*x+d)^n)/e^2-2*b^2*d^2*m*n*ln(-e*x/d)*ln(c*(e*x +d)^n)/e^2+2*b^2*d*n*(e*x+d)*ln(f*x^m)*ln(c*(e*x+d)^n)/e^2+1/2*b*m*n*(e*x+ d)^2*(a+b*ln(c*(e*x+d)^n))/e^2+1/2*b*d^2*m*n*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^ n))/e^2-1/2*b*n*(e*x+d)^2*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))/e^2+1/2*d*m*(e*x +d)*(a+b*ln(c*(e*x+d)^n))^2/e^2-1/4*m*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^ 2+1/2*d^2*m*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))^2/e^2-d*(e*x+d)*ln(f*x^m)*(a+ b*ln(c*(e*x+d)^n))^2/e^2+1/2*(e*x+d)^2*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/e ^2-3/2*b^2*d^2*m*n^2*polylog(2,1+e*x/d)/e^2+b*d^2*m*n*(a+b*ln(c*(e*x+d)^n) )*polylog(2,1+e*x/d)/e^2-b^2*d^2*m*n^2*polylog(3,1+e*x/d)/e^2
Time = 0.17 (sec) , antiderivative size = 825, normalized size of antiderivative = 1.37 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=b^2 n^2 \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (\frac {1}{2} x^2 \log ^2(d+e x)-e \left (\frac {3 d x}{2 e^2}-\frac {x^2}{4 e}-\frac {3 d^2 \log (d+e x)}{2 e^3}-\frac {d x \log (d+e x)}{e^2}+\frac {x^2 \log (d+e x)}{2 e}+\frac {d^2 \log ^2(d+e x)}{2 e^3}\right )\right )+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (\frac {1}{2} x^2 \log (d+e x)-\frac {1}{2} e \left (-\frac {d x}{e^2}+\frac {x^2}{2 e}+\frac {d^2 \log (d+e x)}{e^3}\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )+\frac {1}{2} m x^2 \log (x) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )^2+\frac {1}{4} x^2 \left (-m+2 \left (-m \log (x)+\log \left (f x^m\right )\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )^2+b m n \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right ) \left (-\frac {1}{2} x^2 \log (d+e x)+x^2 \log (x) \log (d+e x)+\frac {1}{2} e \left (-\frac {d x}{e^2}+\frac {x^2}{2 e}+\frac {d^2 \log (d+e x)}{e^3}\right )-e \left (-\frac {d x (-1+\log (x))}{e^2}+\frac {-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)}{e}+\frac {d^2 \left (\frac {\log (x) \log \left (\frac {d+e x}{d}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{e^2}\right )\right )+\frac {1}{2} b^2 m n^2 \left (-\frac {1}{2} x^2 \log ^2(d+e x)+x^2 \log (x) \log ^2(d+e x)+e \left (\frac {3 d x}{2 e^2}-\frac {x^2}{4 e}-\frac {3 d^2 \log (d+e x)}{2 e^3}-\frac {d x \log (d+e x)}{e^2}+\frac {x^2 \log (d+e x)}{2 e}+\frac {d^2 \log ^2(d+e x)}{2 e^3}\right )-2 e \left (-\frac {d \left (2 e x-d \log (d+e x)-e x \log (d+e x)+\log (x) \left (-e x+e x \log (d+e x)+d \log \left (1+\frac {e x}{d}\right )\right )+d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )}{e^3}+\frac {-3 d e x+e^2 x^2+d^2 \log (d+e x)-e^2 x^2 \log (d+e x)+\log (x) \left (e x (2 d-e x)+2 e^2 x^2 \log (d+e x)-2 d^2 \log \left (1+\frac {e x}{d}\right )\right )-2 d^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^3}+\frac {d^2 \left (\frac {1}{2} \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log ^2(d+e x)-\log (d+e x) \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+\operatorname {PolyLog}\left (3,\frac {d+e x}{d}\right )\right )}{e^3}\right )\right ) \]
b^2*n^2*(-(m*Log[x]) + Log[f*x^m])*((x^2*Log[d + e*x]^2)/2 - e*((3*d*x)/(2 *e^2) - x^2/(4*e) - (3*d^2*Log[d + e*x])/(2*e^3) - (d*x*Log[d + e*x])/e^2 + (x^2*Log[d + e*x])/(2*e) + (d^2*Log[d + e*x]^2)/(2*e^3))) + 2*b*n*(-(m*L og[x]) + Log[f*x^m])*((x^2*Log[d + e*x])/2 - (e*(-((d*x)/e^2) + x^2/(2*e) + (d^2*Log[d + e*x])/e^3))/2)*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^ n])) + (m*x^2*Log[x]*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))^2)/2 + (x^2*(-m + 2*(-(m*Log[x]) + Log[f*x^m]))*(a + b*(-(n*Log[d + e*x]) + Lo g[c*(d + e*x)^n]))^2)/4 + b*m*n*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x )^n]))*(-1/2*(x^2*Log[d + e*x]) + x^2*Log[x]*Log[d + e*x] + (e*(-((d*x)/e^ 2) + x^2/(2*e) + (d^2*Log[d + e*x])/e^3))/2 - e*(-((d*x*(-1 + Log[x]))/e^2 ) + (-1/4*x^2 + (x^2*Log[x])/2)/e + (d^2*((Log[x]*Log[(d + e*x)/d])/e + Po lyLog[2, -((e*x)/d)]/e))/e^2)) + (b^2*m*n^2*(-1/2*(x^2*Log[d + e*x]^2) + x ^2*Log[x]*Log[d + e*x]^2 + e*((3*d*x)/(2*e^2) - x^2/(4*e) - (3*d^2*Log[d + e*x])/(2*e^3) - (d*x*Log[d + e*x])/e^2 + (x^2*Log[d + e*x])/(2*e) + (d^2* Log[d + e*x]^2)/(2*e^3)) - 2*e*(-((d*(2*e*x - d*Log[d + e*x] - e*x*Log[d + e*x] + Log[x]*(-(e*x) + e*x*Log[d + e*x] + d*Log[1 + (e*x)/d]) + d*PolyLo g[2, -((e*x)/d)]))/e^3) + (-3*d*e*x + e^2*x^2 + d^2*Log[d + e*x] - e^2*x^2 *Log[d + e*x] + Log[x]*(e*x*(2*d - e*x) + 2*e^2*x^2*Log[d + e*x] - 2*d^2*L og[1 + (e*x)/d]) - 2*d^2*PolyLog[2, -((e*x)/d)])/(4*e^3) + (d^2*(((Log[x] - Log[-((e*x)/d)])*Log[d + e*x]^2)/2 - Log[d + e*x]*PolyLog[2, (d + e*x...
Time = 1.49 (sec) , antiderivative size = 590, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2875, 2009}
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.
| \(\displaystyle \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2875 |
| \(\displaystyle -m \int \left (\frac {n^2 (d+e x)^2 b^2}{4 e^2 x}+\frac {2 d n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2 x}+\frac {2 d n (a-b n) b}{e}-\frac {n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{2 e^2 x}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 x}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 x}\right )dx-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -m \left (-\frac {b d^2 n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b d^2 n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {d^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {a b d n x}{2 e}+\frac {2 b d n x (a-b n)}{e}+\frac {2 b^2 d^2 n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {5 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}+\frac {3 b^2 d^2 n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{2 e^2}+\frac {b^2 d^2 n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e^2}+\frac {b^2 d^2 n^2 \log (x)}{4 e^2}+\frac {b^2 n^2 (d+e x)^2}{4 e^2}-\frac {2 b^2 d n^2 x}{e}+\frac {1}{8} b^2 n^2 x^2\right )-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}\) |
(2*a*b*d*n*x*Log[f*x^m])/e - (2*b^2*d*n^2*x*Log[f*x^m])/e + (b^2*n^2*(d + e*x)^2*Log[f*x^m])/(4*e^2) + (2*b^2*d*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e* x)^n])/e^2 - (b*n*(d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(2*e^ 2) - (d*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/e^2 + ((d + e*x )^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) - m*((a*b*d*n*x)/(2*e ) - (2*b^2*d*n^2*x)/e + (2*b*d*n*(a - b*n)*x)/e + (b^2*n^2*x^2)/8 + (b^2*n ^2*(d + e*x)^2)/(4*e^2) + (b^2*d^2*n^2*Log[x])/(4*e^2) + (5*b^2*d*n*(d + e *x)*Log[c*(d + e*x)^n])/(2*e^2) + (2*b^2*d^2*n*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e^2 - (b*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) - (b*d ^2*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) - (d*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) + ((d + e*x)^2*(a + b*Log[c*(d + e*x)^ n])^2)/(4*e^2) - (d^2*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2 ) + (3*b^2*d^2*n^2*PolyLog[2, 1 + (e*x)/d])/(2*e^2) - (b*d^2*n*(a + b*Log[ c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e^2 + (b^2*d^2*n^2*PolyLog[3, 1 + (e*x)/d])/e^2)
3.4.68.3.1 Defintions of rubi rules used
Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_ .))^(p_)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b* Log[c*(d + e*x)^n])^p, x]}, Simp[Log[f*x^m] u, x] - Simp[m Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 1] && IGtQ[ q, 0]
\[\int x \ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}d x\]
\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]
integral(b^2*x*log((e*x + d)^n*c)^2*log(f*x^m) + 2*a*b*x*log((e*x + d)^n*c )*log(f*x^m) + a^2*x*log(f*x^m), x)
Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]
-1/4*(b^2*(m - 2*log(f))*x^2 - 2*b^2*x^2*log(x^m))*log((e*x + d)^n)^2 + in tegrate(1/2*(2*(b^2*e*log(c)^2*log(f) + 2*a*b*e*log(c)*log(f) + a^2*e*log( f))*x^2 + 2*(b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f)) *x + ((4*a*b*e*log(f) + (4*e*log(c)*log(f) + (m*n - 2*n*log(f))*e)*b^2)*x^ 2 + 4*(b^2*d*log(c)*log(f) + a*b*d*log(f))*x - 2*(((e*n - 2*e*log(c))*b^2 - 2*a*b*e)*x^2 - 2*(b^2*d*log(c) + a*b*d)*x)*log(x^m))*log((e*x + d)^n) + 2*((b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^2 + (b^2*d*log(c)^2 + 2*a*b *d*log(c) + a^2*d)*x)*log(x^m))/(e*x + d), x)
\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]
Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int x\,\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \]