Integrand size = 24, antiderivative size = 283 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=-\frac {5 b e^3 m n x}{16 f^3}+\frac {3 b e^2 m n x^2}{32 f^2}-\frac {7 b e m n x^3}{144 f}+\frac {1}{32} b m n x^4+\frac {e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac {1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 m n \log (e+f x)}{16 f^4}+\frac {b e^4 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac {e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}-\frac {1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {b e^4 m n \operatorname {PolyLog}\left (2,1+\frac {f x}{e}\right )}{4 f^4} \]
-5/16*b*e^3*m*n*x/f^3+3/32*b*e^2*m*n*x^2/f^2-7/144*b*e*m*n*x^3/f+1/32*b*m* n*x^4+1/4*e^3*m*x*(a+b*ln(c*x^n))/f^3-1/8*e^2*m*x^2*(a+b*ln(c*x^n))/f^2+1/ 12*e*m*x^3*(a+b*ln(c*x^n))/f-1/16*m*x^4*(a+b*ln(c*x^n))+1/16*b*e^4*m*n*ln( f*x+e)/f^4+1/4*b*e^4*m*n*ln(-f*x/e)*ln(f*x+e)/f^4-1/4*e^4*m*(a+b*ln(c*x^n) )*ln(f*x+e)/f^4-1/16*b*n*x^4*ln(d*(f*x+e)^m)+1/4*x^4*(a+b*ln(c*x^n))*ln(d* (f*x+e)^m)+1/4*b*e^4*m*n*polylog(2,1+f*x/e)/f^4
Time = 0.17 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.02 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=-\frac {-72 a e^3 f m x+90 b e^3 f m n x+36 a e^2 f^2 m x^2-27 b e^2 f^2 m n x^2-24 a e f^3 m x^3+14 b e f^3 m n x^3+18 a f^4 m x^4-9 b f^4 m n x^4+72 a e^4 m \log (e+f x)-18 b e^4 m n \log (e+f x)-72 b e^4 m n \log (x) \log (e+f x)-72 a f^4 x^4 \log \left (d (e+f x)^m\right )+18 b f^4 n x^4 \log \left (d (e+f x)^m\right )+6 b \log \left (c x^n\right ) \left (f m x \left (-12 e^3+6 e^2 f x-4 e f^2 x^2+3 f^3 x^3\right )+12 e^4 m \log (e+f x)-12 f^4 x^4 \log \left (d (e+f x)^m\right )\right )+72 b e^4 m n \log (x) \log \left (1+\frac {f x}{e}\right )+72 b e^4 m n \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{288 f^4} \]
-1/288*(-72*a*e^3*f*m*x + 90*b*e^3*f*m*n*x + 36*a*e^2*f^2*m*x^2 - 27*b*e^2 *f^2*m*n*x^2 - 24*a*e*f^3*m*x^3 + 14*b*e*f^3*m*n*x^3 + 18*a*f^4*m*x^4 - 9* b*f^4*m*n*x^4 + 72*a*e^4*m*Log[e + f*x] - 18*b*e^4*m*n*Log[e + f*x] - 72*b *e^4*m*n*Log[x]*Log[e + f*x] - 72*a*f^4*x^4*Log[d*(e + f*x)^m] + 18*b*f^4* n*x^4*Log[d*(e + f*x)^m] + 6*b*Log[c*x^n]*(f*m*x*(-12*e^3 + 6*e^2*f*x - 4* e*f^2*x^2 + 3*f^3*x^3) + 12*e^4*m*Log[e + f*x] - 12*f^4*x^4*Log[d*(e + f*x )^m]) + 72*b*e^4*m*n*Log[x]*Log[1 + (f*x)/e] + 72*b*e^4*m*n*PolyLog[2, -(( f*x)/e)])/f^4
Time = 0.45 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2823, 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.
| \(\displaystyle \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (-\frac {m \log (e+f x) e^4}{4 f^4 x}+\frac {m e^3}{4 f^3}-\frac {m x e^2}{8 f^2}+\frac {m x^2 e}{12 f}-\frac {m x^3}{16}+\frac {1}{4} x^3 \log \left (d (e+f x)^m\right )\right )dx+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {e^4 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac {1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {e^4 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac {1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {1}{16} x^4 \log \left (d (e+f x)^m\right )-\frac {e^4 m \operatorname {PolyLog}\left (2,\frac {f x}{e}+1\right )}{4 f^4}-\frac {e^4 m \log (e+f x)}{16 f^4}-\frac {e^4 m \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{4 f^4}+\frac {5 e^3 m x}{16 f^3}-\frac {3 e^2 m x^2}{32 f^2}+\frac {7 e m x^3}{144 f}-\frac {m x^4}{32}\right )\) |
(e^3*m*x*(a + b*Log[c*x^n]))/(4*f^3) - (e^2*m*x^2*(a + b*Log[c*x^n]))/(8*f ^2) + (e*m*x^3*(a + b*Log[c*x^n]))/(12*f) - (m*x^4*(a + b*Log[c*x^n]))/16 - (e^4*m*(a + b*Log[c*x^n])*Log[e + f*x])/(4*f^4) + (x^4*(a + b*Log[c*x^n] )*Log[d*(e + f*x)^m])/4 - b*n*((5*e^3*m*x)/(16*f^3) - (3*e^2*m*x^2)/(32*f^ 2) + (7*e*m*x^3)/(144*f) - (m*x^4)/32 - (e^4*m*Log[e + f*x])/(16*f^4) - (e ^4*m*Log[-((f*x)/e)]*Log[e + f*x])/(4*f^4) + (x^4*Log[d*(e + f*x)^m])/16 - (e^4*m*PolyLog[2, 1 + (f*x)/e])/(4*f^4))
3.1.70.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 85.59 (sec) , antiderivative size = 1248, normalized size of antiderivative = 4.41
-1/16*x^4*a*m-205/576*b*e^4*m*n/f^4+(1/4*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*( f*x+e)^m)^2-1/4*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*csgn(I*d)-1/4*I *Pi*csgn(I*d*(f*x+e)^m)^3+1/4*I*Pi*csgn(I*d*(f*x+e)^m)^2*csgn(I*d)+1/2*ln( d))*(1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csg n(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b *ln(c)+2*a)*x^4+1/2*b*x^4*ln(x^n)-1/8*b*n*x^4)+(1/4*b*x^4*ln(x^n)+1/16*x^4 *(-2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*I*b*Pi*csgn(I*c)*csgn(I* c*x^n)^2+2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*b*Pi*csgn(I*c*x^n)^3+4*b *ln(c)-b*n+4*a))*ln((f*x+e)^m)-1/32*I*m*x^4*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2 -1/32*I*m*x^4*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*I*m/f^4*e^4*ln(f*x+e)*b *Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/8*I*m/f^4*e^4*ln(f*x+e)*b*Pi*csgn(I*x^n)*c sgn(I*c*x^n)^2-1/24*I*m/f*e*x^3*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1 /16*I*m/f^2*x^2*e^2*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/8*I*m/f^3*x *e^3*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*I*m/f^4*e^4*ln(f*x+e)*b* Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/24*I*m/f*e*x^3*b*Pi*csgn(I*c)*csg n(I*c*x^n)^2+1/24*I*m/f*e*x^3*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/16*I*m/f^ 2*x^2*e^2*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/16*I*m/f^2*x^2*e^2*b*Pi*csgn(I* x^n)*csgn(I*c*x^n)^2+1/8*I*m/f^3*x*e^3*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/8* I*m/f^3*x*e^3*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*n*b/f^4*e^4*m*dilog(-f* x/e)-1/16*m*x^4*b*ln(c)-1/16*m*b*ln(x^n)*x^4+1/32*I*m*x^4*b*Pi*csgn(I*c...
\[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{3} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.35 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=-\frac {{\left (\log \left (\frac {f x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {f x}{e}\right )\right )} b e^{4} m n}{4 \, f^{4}} - \frac {{\left (4 \, a e^{4} m - {\left (e^{4} m n - 4 \, e^{4} m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{16 \, f^{4}} + \frac {72 \, b e^{4} m n \log \left (f x + e\right ) \log \left (x\right ) - 9 \, {\left (2 \, {\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} a - {\left (f^{4} m n - 2 \, f^{4} n \log \left (d\right ) - 2 \, {\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{4} + 2 \, {\left (12 \, a e f^{3} m - {\left (7 \, e f^{3} m n - 12 \, e f^{3} m \log \left (c\right )\right )} b\right )} x^{3} - 9 \, {\left (4 \, a e^{2} f^{2} m - {\left (3 \, e^{2} f^{2} m n - 4 \, e^{2} f^{2} m \log \left (c\right )\right )} b\right )} x^{2} + 18 \, {\left (4 \, a e^{3} f m - {\left (5 \, e^{3} f m n - 4 \, e^{3} f m \log \left (c\right )\right )} b\right )} x + 18 \, {\left (4 \, b f^{4} x^{4} \log \left (x^{n}\right ) + {\left (4 \, a f^{4} - {\left (f^{4} n - 4 \, f^{4} \log \left (c\right )\right )} b\right )} x^{4}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) + 6 \, {\left (4 \, b e f^{3} m x^{3} - 6 \, b e^{2} f^{2} m x^{2} + 12 \, b e^{3} f m x - 12 \, b e^{4} m \log \left (f x + e\right ) - 3 \, {\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} b x^{4}\right )} \log \left (x^{n}\right )}{288 \, f^{4}} \]
-1/4*(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*e^4*m*n/f^4 - 1/16*(4*a*e^4 *m - (e^4*m*n - 4*e^4*m*log(c))*b)*log(f*x + e)/f^4 + 1/288*(72*b*e^4*m*n* log(f*x + e)*log(x) - 9*(2*(f^4*m - 4*f^4*log(d))*a - (f^4*m*n - 2*f^4*n*l og(d) - 2*(f^4*m - 4*f^4*log(d))*log(c))*b)*x^4 + 2*(12*a*e*f^3*m - (7*e*f ^3*m*n - 12*e*f^3*m*log(c))*b)*x^3 - 9*(4*a*e^2*f^2*m - (3*e^2*f^2*m*n - 4 *e^2*f^2*m*log(c))*b)*x^2 + 18*(4*a*e^3*f*m - (5*e^3*f*m*n - 4*e^3*f*m*log (c))*b)*x + 18*(4*b*f^4*x^4*log(x^n) + (4*a*f^4 - (f^4*n - 4*f^4*log(c))*b )*x^4)*log((f*x + e)^m) + 6*(4*b*e*f^3*m*x^3 - 6*b*e^2*f^2*m*x^2 + 12*b*e^ 3*f*m*x - 12*b*e^4*m*log(f*x + e) - 3*(f^4*m - 4*f^4*log(d))*b*x^4)*log(x^ n))/f^4
\[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{3} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\int x^3\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]