Integrand size = 23, antiderivative size = 473 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=-12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-18 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )}{f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{f} \]
-12*a*b^2*m*n^2*x+18*b^3*m*n^3*x-6*b^2*m*n^2*(-b*n+a)*x-18*b^3*m*n^2*x*ln( c*x^n)+6*b*m*n*x*(a+b*ln(c*x^n))^2-m*x*(a+b*ln(c*x^n))^3+6*b^2*e*m*n^2*(-b *n+a)*ln(f*x+e)/f+6*a*b^2*n^2*x*ln(d*(f*x+e)^m)-6*b^3*n^3*x*ln(d*(f*x+e)^m )+6*b^3*n^2*x*ln(c*x^n)*ln(d*(f*x+e)^m)-3*b*n*x*(a+b*ln(c*x^n))^2*ln(d*(f* x+e)^m)+x*(a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)+6*b^3*e*m*n^2*ln(c*x^n)*ln(1+f *x/e)/f-3*b*e*m*n*(a+b*ln(c*x^n))^2*ln(1+f*x/e)/f+e*m*(a+b*ln(c*x^n))^3*ln (1+f*x/e)/f+6*b^3*e*m*n^3*polylog(2,-f*x/e)/f-6*b^2*e*m*n^2*(a+b*ln(c*x^n) )*polylog(2,-f*x/e)/f+3*b*e*m*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x/e)/f+6*b^ 3*e*m*n^3*polylog(3,-f*x/e)/f-6*b^2*e*m*n^2*(a+b*ln(c*x^n))*polylog(3,-f*x /e)/f+6*b^3*e*m*n^3*polylog(4,-f*x/e)/f
Leaf count is larger than twice the leaf count of optimal. \(1122\) vs. \(2(473)=946\).
Time = 0.25 (sec) , antiderivative size = 1122, normalized size of antiderivative = 2.37 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\frac {-a^3 f m x+6 a^2 b f m n x-18 a b^2 f m n^2 x+24 b^3 f m n^3 x-3 a^2 b f m x \log \left (c x^n\right )+12 a b^2 f m n x \log \left (c x^n\right )-18 b^3 f m n^2 x \log \left (c x^n\right )-3 a b^2 f m x \log ^2\left (c x^n\right )+6 b^3 f m n x \log ^2\left (c x^n\right )-b^3 f m x \log ^3\left (c x^n\right )+a^3 e m \log (e+f x)-3 a^2 b e m n \log (e+f x)+6 a b^2 e m n^2 \log (e+f x)-6 b^3 e m n^3 \log (e+f x)-3 a^2 b e m n \log (x) \log (e+f x)+6 a b^2 e m n^2 \log (x) \log (e+f x)-6 b^3 e m n^3 \log (x) \log (e+f x)+3 a b^2 e m n^2 \log ^2(x) \log (e+f x)-3 b^3 e m n^3 \log ^2(x) \log (e+f x)-b^3 e m n^3 \log ^3(x) \log (e+f x)+3 a^2 b e m \log \left (c x^n\right ) \log (e+f x)-6 a b^2 e m n \log \left (c x^n\right ) \log (e+f x)+6 b^3 e m n^2 \log \left (c x^n\right ) \log (e+f x)-6 a b^2 e m n \log (x) \log \left (c x^n\right ) \log (e+f x)+6 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log (e+f x)+3 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log (e+f x)+3 a b^2 e m \log ^2\left (c x^n\right ) \log (e+f x)-3 b^3 e m n \log ^2\left (c x^n\right ) \log (e+f x)-3 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log (e+f x)+b^3 e m \log ^3\left (c x^n\right ) \log (e+f x)+a^3 f x \log \left (d (e+f x)^m\right )-3 a^2 b f n x \log \left (d (e+f x)^m\right )+6 a b^2 f n^2 x \log \left (d (e+f x)^m\right )-6 b^3 f n^3 x \log \left (d (e+f x)^m\right )+3 a^2 b f x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 a b^2 f n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^3 f n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 f x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b^3 f n x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 f x \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a^2 b e m n \log (x) \log \left (1+\frac {f x}{e}\right )-6 a b^2 e m n^2 \log (x) \log \left (1+\frac {f x}{e}\right )+6 b^3 e m n^3 \log (x) \log \left (1+\frac {f x}{e}\right )-3 a b^2 e m n^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )+3 b^3 e m n^3 \log ^2(x) \log \left (1+\frac {f x}{e}\right )+b^3 e m n^3 \log ^3(x) \log \left (1+\frac {f x}{e}\right )+6 a b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-6 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-3 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+3 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+3 b e m n \left (a^2-2 a b n+2 b^2 n^2+2 b (a-b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )-6 b^2 e m n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )+6 b^3 e m n^3 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{f} \]
(-(a^3*f*m*x) + 6*a^2*b*f*m*n*x - 18*a*b^2*f*m*n^2*x + 24*b^3*f*m*n^3*x - 3*a^2*b*f*m*x*Log[c*x^n] + 12*a*b^2*f*m*n*x*Log[c*x^n] - 18*b^3*f*m*n^2*x* Log[c*x^n] - 3*a*b^2*f*m*x*Log[c*x^n]^2 + 6*b^3*f*m*n*x*Log[c*x^n]^2 - b^3 *f*m*x*Log[c*x^n]^3 + a^3*e*m*Log[e + f*x] - 3*a^2*b*e*m*n*Log[e + f*x] + 6*a*b^2*e*m*n^2*Log[e + f*x] - 6*b^3*e*m*n^3*Log[e + f*x] - 3*a^2*b*e*m*n* Log[x]*Log[e + f*x] + 6*a*b^2*e*m*n^2*Log[x]*Log[e + f*x] - 6*b^3*e*m*n^3* Log[x]*Log[e + f*x] + 3*a*b^2*e*m*n^2*Log[x]^2*Log[e + f*x] - 3*b^3*e*m*n^ 3*Log[x]^2*Log[e + f*x] - b^3*e*m*n^3*Log[x]^3*Log[e + f*x] + 3*a^2*b*e*m* Log[c*x^n]*Log[e + f*x] - 6*a*b^2*e*m*n*Log[c*x^n]*Log[e + f*x] + 6*b^3*e* m*n^2*Log[c*x^n]*Log[e + f*x] - 6*a*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[e + f* x] + 6*b^3*e*m*n^2*Log[x]*Log[c*x^n]*Log[e + f*x] + 3*b^3*e*m*n^2*Log[x]^2 *Log[c*x^n]*Log[e + f*x] + 3*a*b^2*e*m*Log[c*x^n]^2*Log[e + f*x] - 3*b^3*e *m*n*Log[c*x^n]^2*Log[e + f*x] - 3*b^3*e*m*n*Log[x]*Log[c*x^n]^2*Log[e + f *x] + b^3*e*m*Log[c*x^n]^3*Log[e + f*x] + a^3*f*x*Log[d*(e + f*x)^m] - 3*a ^2*b*f*n*x*Log[d*(e + f*x)^m] + 6*a*b^2*f*n^2*x*Log[d*(e + f*x)^m] - 6*b^3 *f*n^3*x*Log[d*(e + f*x)^m] + 3*a^2*b*f*x*Log[c*x^n]*Log[d*(e + f*x)^m] - 6*a*b^2*f*n*x*Log[c*x^n]*Log[d*(e + f*x)^m] + 6*b^3*f*n^2*x*Log[c*x^n]*Log [d*(e + f*x)^m] + 3*a*b^2*f*x*Log[c*x^n]^2*Log[d*(e + f*x)^m] - 3*b^3*f*n* x*Log[c*x^n]^2*Log[d*(e + f*x)^m] + b^3*f*x*Log[c*x^n]^3*Log[d*(e + f*x)^m ] + 3*a^2*b*e*m*n*Log[x]*Log[1 + (f*x)/e] - 6*a*b^2*e*m*n^2*Log[x]*Log[...
Time = 0.86 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2818, 6, 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.
| \(\displaystyle \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx\) |
| \(\Big \downarrow \) 2818 |
| \(\displaystyle -f m \int \left (\frac {6 n^2 x \log \left (c x^n\right ) b^3}{e+f x}-\frac {6 n^3 x b^3}{e+f x}+\frac {6 a n^2 x b^2}{e+f x}-\frac {3 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{e+f x}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x}\right )dx+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -f m \int \left (\frac {6 n^2 x \log \left (c x^n\right ) b^3}{e+f x}-\frac {3 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{e+f x}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x}+\frac {\left (6 a b^2 n^2-6 b^3 n^3\right ) x}{e+f x}\right )dx+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-f m \left (\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac {6 b^2 e n^2 (a-b n) \log (e+f x)}{f^2}+\frac {12 a b^2 n^2 x}{f}+\frac {6 b^2 n^2 x (a-b n)}{f}-\frac {3 b e n \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {3 b e n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac {e \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}-\frac {6 b n x \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {6 b^3 e n^2 \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )}{f^2}+\frac {18 b^3 n^2 x \log \left (c x^n\right )}{f}-\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{f^2}-\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{f^2}-\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{f^2}-\frac {18 b^3 n^3 x}{f}\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )\) |
6*a*b^2*n^2*x*Log[d*(e + f*x)^m] - 6*b^3*n^3*x*Log[d*(e + f*x)^m] + 6*b^3* n^2*x*Log[c*x^n]*Log[d*(e + f*x)^m] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[d*( e + f*x)^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m] - f*m*((12*a*b^2*n ^2*x)/f - (18*b^3*n^3*x)/f + (6*b^2*n^2*(a - b*n)*x)/f + (18*b^3*n^2*x*Log [c*x^n])/f - (6*b*n*x*(a + b*Log[c*x^n])^2)/f + (x*(a + b*Log[c*x^n])^3)/f - (6*b^2*e*n^2*(a - b*n)*Log[e + f*x])/f^2 - (6*b^3*e*n^2*Log[c*x^n]*Log[ 1 + (f*x)/e])/f^2 + (3*b*e*n*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/f^2 - (e*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/f^2 - (6*b^3*e*n^3*PolyLog[2, -( (f*x)/e)])/f^2 + (6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f ^2 - (3*b*e*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x)/e)])/f^2 - (6*b^3*e* n^3*PolyLog[3, -((f*x)/e)])/f^2 + (6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[ 3, -((f*x)/e)])/f^2 - (6*b^3*e*n^3*PolyLog[4, -((f*x)/e)])/f^2)
3.1.86.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r] u, x] - Simp[f*m*r Int[x^(m - 1)/(e + f*x^m) u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && Inte gerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 159.67 (sec) , antiderivative size = 15386, normalized size of antiderivative = 32.53
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
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)
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\text {Timed out} \]
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
((b^3*e*m*log(f*x + e) - (f*m - f*log(d))*b^3*x)*log(x^n)^3 + (b^3*f*x*log (x^n)^3 - 3*((f*n - f*log(c))*b^3 - a*b^2*f)*x*log(x^n)^2 - 3*(2*(f*n - f* log(c))*a*b^2 - (2*f*n^2 - 2*f*n*log(c) + f*log(c)^2)*b^3 - a^2*b*f)*x*log (x^n) - (3*(f*n - f*log(c))*a^2*b - 3*(2*f*n^2 - 2*f*n*log(c) + f*log(c)^2 )*a*b^2 + (6*f*n^3 - 6*f*n^2*log(c) + 3*f*n*log(c)^2 - f*log(c)^3)*b^3 - a ^3*f)*x)*log((f*x + e)^m))/f - integrate((((f^2*m - f^2*log(d))*a^3 - 3*(f ^2*m*n - (f^2*m - f^2*log(d))*log(c))*a^2*b + 3*(2*f^2*m*n^2 - 2*f^2*m*n*l og(c) + (f^2*m - f^2*log(d))*log(c)^2)*a*b^2 - (6*f^2*m*n^3 - 6*f^2*m*n^2* log(c) + 3*f^2*m*n*log(c)^2 - (f^2*m - f^2*log(d))*log(c)^3)*b^3)*x^2 + 3* (((f^2*m - f^2*log(d))*a*b^2 - (2*f^2*m*n - f^2*n*log(d) - (f^2*m - f^2*lo g(d))*log(c))*b^3)*x^2 - (a*b^2*e*f*log(d) + (e*f*m*n - e*f*n*log(d) + e*f *log(c)*log(d))*b^3)*x + (b^3*e*f*m*n*x + b^3*e^2*m*n)*log(f*x + e))*log(x ^n)^2 - (b^3*e*f*log(c)^3*log(d) + 3*a*b^2*e*f*log(c)^2*log(d) + 3*a^2*b*e *f*log(c)*log(d) + a^3*e*f*log(d))*x + 3*(((f^2*m - f^2*log(d))*a^2*b - 2* (f^2*m*n - (f^2*m - f^2*log(d))*log(c))*a*b^2 + (2*f^2*m*n^2 - 2*f^2*m*n*l og(c) + (f^2*m - f^2*log(d))*log(c)^2)*b^3)*x^2 - (b^3*e*f*log(c)^2*log(d) + 2*a*b^2*e*f*log(c)*log(d) + a^2*b*e*f*log(d))*x)*log(x^n))/(f^2*x^2 + e *f*x), x)
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int \ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]