Integrand size = 23, antiderivative size = 522 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=-12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e}-\frac {3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )}{e} \]
[Out]
Time = 0.61 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2436, 2333, 2332, 2470, 2458, 45, 2393, 2354, 2438, 2395, 2421, 6724, 2430} \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {6 b^2 d m n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {6 b^2 d m n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+6 a b^2 n^2 x \log \left (f x^m\right )-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {3 b d m n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \operatorname {PolyLog}\left (4,\frac {e x}{d}+1\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )+18 b^3 m n^3 x \]
[In]
[Out]
Rule 45
Rule 2332
Rule 2333
Rule 2354
Rule 2393
Rule 2395
Rule 2421
Rule 2430
Rule 2436
Rule 2438
Rule 2458
Rule 2470
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-m \int \left (6 a b^2 n^2-6 b^3 n^3+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e x}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e x}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e x}\right ) \, dx \\ & = -6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {m \int \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx}{e}+\frac {(3 b m n) \int \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{e}-\frac {\left (6 b^3 m n^2\right ) \int \frac {(d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{e} \\ & = -6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {m \text {Subst}\left (\int \frac {x \left (a+b \log \left (c x^n\right )\right )^3}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{e^2}+\frac {(3 b m n) \text {Subst}\left (\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{e^2}-\frac {\left (6 b^3 m n^2\right ) \text {Subst}\left (\int \frac {x \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{e^2} \\ & = -6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {m \text {Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^3-\frac {d e \left (a+b \log \left (c x^n\right )\right )^3}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}+\frac {(3 b m n) \text {Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^2-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}-\frac {\left (6 b^3 m n^2\right ) \text {Subst}\left (\int \left (e \log \left (c x^n\right )-\frac {d e \log \left (c x^n\right )}{d-x}\right ) \, dx,x,d+e x\right )}{e^2} \\ & = -6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {m \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}+\frac {(d m) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d-x} \, dx,x,d+e x\right )}{e}+\frac {(3 b m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-\frac {(3 b d m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^3 m n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac {\left (6 b^3 d m n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{e} \\ & = 6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {6 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {3 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(3 b m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}+\frac {(3 b d m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^2 m n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^2 d m n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac {\left (6 b^3 d m n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -6 a b^2 m n^2 x+6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {6 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {6 b^3 d m n^3 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {\left (6 b^2 m n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^3 m n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac {\left (6 b^2 d m n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^3 d m n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -12 a b^2 m n^2 x+12 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {12 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {6 b^3 d m n^3 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}-\frac {\left (6 b^3 m n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^3 d m n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -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+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {6 b^3 d m n^3 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_4\left (1+\frac {e x}{d}\right )}{e} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1163\) vs. \(2(522)=1044\).
Time = 0.46 (sec) , antiderivative size = 1163, normalized size of antiderivative = 2.23 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {-b^3 n^3 (d+e x) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (-6+6 \log (d+e x)-3 \log ^2(d+e x)+\log ^3(d+e x)\right )-3 b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-3 b e n x \left (m-\log \left (f x^m\right )\right ) \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-3 b d n \left (m+m \log (x)-\log \left (f x^m\right )\right ) \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+e x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (3 b m n+3 b n \left (m \log (x)-\log \left (f x^m\right )\right )+a \left (-m \log (x)+\log \left (f x^m\right )\right )+b \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )+a d m \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-b d m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-a m \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (e x+\log (x) \left (-e x+d \log \left (1+\frac {e x}{d}\right )\right )+d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+3 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (e x+\log (x) \left (-e x+d \log \left (1+\frac {e x}{d}\right )\right )+d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+b m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (e x+\log (x) \left (-e x+d \log \left (1+\frac {e x}{d}\right )\right )+d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-3 b^2 m n^2 \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (-6 e x+2 e x \log (x)+4 d \log (d+e x)+4 e x \log (d+e x)-2 e x \log (x) \log (d+e x)-d \log ^2(d+e x)-e x \log ^2(d+e x)+d \log (x) \log ^2(d+e x)+e x \log (x) \log ^2(d+e x)-d \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-2 d \log (x) \log \left (1+\frac {e x}{d}\right )-2 d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 d \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+2 d \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )+b^3 m n^3 \left (6 d+24 e x-6 e x \log (x)-18 d \log (d+e x)-18 e x \log (d+e x)+6 e x \log (x) \log (d+e x)+6 d \log ^2(d+e x)+6 e x \log ^2(d+e x)-3 d \log (x) \log ^2(d+e x)-3 e x \log (x) \log ^2(d+e x)+3 d \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-d \log ^3(d+e x)-e x \log ^3(d+e x)+d \log (x) \log ^3(d+e x)+e x \log (x) \log ^3(d+e x)-d \log \left (-\frac {e x}{d}\right ) \log ^3(d+e x)+6 d \log (x) \log \left (1+\frac {e x}{d}\right )+6 d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-3 d (-2+\log (d+e x)) \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-6 d \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )+6 d \log (d+e x) \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )-6 d \operatorname {PolyLog}\left (4,1+\frac {e x}{d}\right )\right )}{e} \]
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\[\int \ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}d x\]
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\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right ) \,d x } \]
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Timed out. \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\text {Timed out} \]
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\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right ) \,d x } \]
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\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right ) \,d x } \]
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Timed out. \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\int \ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3 \,d x \]
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