Optimal. Leaf size=522 \[ \frac {6 b^2 d m n^2 \text {Li}_2\left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {6 b^2 d m n^2 \text {Li}_3\left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+6 a b^2 n^2 x \log \left (f x^m\right )-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {3 b d m n \text {Li}_2\left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_2\left (\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_3\left (\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \text {Li}_4\left (\frac {e x}{d}+1\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )+18 b^3 m n^3 x \]
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Rubi [A] time = 0.86, 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 = {2389, 2296, 2295, 2423, 2411, 43, 2351, 2317, 2391, 2353, 2374, 6589, 2383} \[ \frac {6 b^2 d m n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {6 b^2 d m n^2 \text {PolyLog}\left (3,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {3 b d m n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {6 b^3 d m n^3 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \text {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e}-\frac {6 b^3 d m n^3 \text {PolyLog}\left (4,\frac {e x}{d}+1\right )}{e}+6 a b^2 n^2 x \log \left (f x^m\right )-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-\frac {3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b d m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 d m n^2 \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )+18 b^3 m n^3 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2296
Rule 2317
Rule 2351
Rule 2353
Rule 2374
Rule 2383
Rule 2389
Rule 2391
Rule 2411
Rule 2423
Rule 6589
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 \operatorname {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) \operatorname {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 ) \operatorname {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 \operatorname {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) \operatorname {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 ) \operatorname {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 \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}+\frac {(d m) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.54, size = 0, normalized size = 0.00 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} \log \left (f x^{m}\right ) + 3 \, a b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} \log \left (f x^{m}\right ) + 3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a^{3} \log \left (f x^{m}\right ), x\right ) \]
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 \[ \int {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} \log \left (f x^{m}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.19, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{3} \ln \left (f \,x^{m}\right )\, dx \]
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 \[ -{\left (b^{3} {\left (m - \log \relax (f)\right )} x - b^{3} x \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + \int \frac {b^{3} d \log \relax (c)^{3} \log \relax (f) + 3 \, a b^{2} d \log \relax (c)^{2} \log \relax (f) + 3 \, a^{2} b d \log \relax (c) \log \relax (f) + a^{3} d \log \relax (f) + 3 \, {\left (b^{3} d \log \relax (c) \log \relax (f) + a b^{2} d \log \relax (f) + {\left (a b^{2} e \log \relax (f) + {\left (e \log \relax (c) \log \relax (f) + {\left (m n - n \log \relax (f)\right )} e\right )} b^{3}\right )} x + {\left (b^{3} d \log \relax (c) + a b^{2} d - {\left ({\left (e n - e \log \relax (c)\right )} b^{3} - a b^{2} e\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + {\left (b^{3} e \log \relax (c)^{3} \log \relax (f) + 3 \, a b^{2} e \log \relax (c)^{2} \log \relax (f) + 3 \, a^{2} b e \log \relax (c) \log \relax (f) + a^{3} e \log \relax (f)\right )} x + 3 \, {\left (b^{3} d \log \relax (c)^{2} \log \relax (f) + 2 \, a b^{2} d \log \relax (c) \log \relax (f) + a^{2} b d \log \relax (f) + {\left (b^{3} e \log \relax (c)^{2} \log \relax (f) + 2 \, a b^{2} e \log \relax (c) \log \relax (f) + a^{2} b e \log \relax (f)\right )} x + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) + {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d + {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x\right )} \log \left (x^{m}\right )}{e x + d}\,{d x} \]
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 \[ \int \ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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