Optimal. Leaf size=522 \[ -12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {6 b^3 d m n^3 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}+\frac {6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_4\left (1+\frac {e x}{d}\right )}{e} \]
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Rubi [A]
time = 0.58, antiderivative size = 522, normalized size of antiderivative = 1.00, number
of steps used = 28, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules
used = {2436, 2333, 2332, 2470, 2458, 45, 2393, 2354, 2438, 2395, 2421, 6724, 2430}
\begin {gather*} \frac {6 b^2 d m n^2 \text {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 \text {PolyLog}\left (3,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {3 b d m n \text {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^3 d m n^3 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \text {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \text {PolyLog}\left (4,\frac {e x}{d}+1\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 {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}-6 b^3 n^3 x \log \left (f x^m\right )+18 b^3 m n^3 x \end {gather*}
Antiderivative was successfully verified.
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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*} \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=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*}
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Mathematica [F]
time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \ln \left (f \,x^{m}\right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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