Optimal. Leaf size=602 \[ \frac {b d^2 m n \text {Li}_2\left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {a b d m n x}{2 e}-\frac {2 b d m n x (a-b n)}{e}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {3 b^2 d^2 m n^2 \text {Li}_2\left (\frac {e x}{d}+1\right )}{2 e^2}-\frac {b^2 d^2 m n^2 \text {Li}_3\left (\frac {e x}{d}+1\right )}{e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {2 b^2 d m n^2 x}{e}-\frac {1}{8} b^2 m n^2 x^2 \]
[Out]
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Rubi [A] time = 1.29, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 16, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304, 2428, 43, 2411, 2351, 2317, 2391, 2353, 2374, 6589} \[ \frac {b d^2 m n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {3 b^2 d^2 m n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{2 e^2}-\frac {b^2 d^2 m n^2 \text {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {a b d m n x}{2 e}-\frac {2 b d m n x (a-b n)}{e}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {2 b^2 d m n^2 x}{e}-\frac {1}{8} b^2 m n^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2317
Rule 2351
Rule 2353
Rule 2374
Rule 2389
Rule 2390
Rule 2391
Rule 2401
Rule 2411
Rule 2428
Rule 6589
Rubi steps
\begin {align*} \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-m \int \left (\frac {2 a b d n}{e}-\frac {2 b^2 d n^2}{e}+\frac {b^2 n^2 (d+e x)^2}{4 e^2 x}+\frac {2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 x}-\frac {b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 x}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 x}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 x}\right ) \, dx\\ &=-\frac {2 b d m n (a-b n) x}{e}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m \int \frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac {(d m) \int \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{e^2}+\frac {(b m n) \int \frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx}{2 e^2}-\frac {\left (2 b^2 d m n\right ) \int \frac {(d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{e^2}-\frac {\left (b^2 m n^2\right ) \int \frac {(d+e x)^2}{x} \, dx}{4 e^2}\\ &=-\frac {2 b d m n (a-b n) x}{e}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 e^3}+\frac {(d m) \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^3}+\frac {(b m n) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 e^3}-\frac {\left (2 b^2 d m n\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^3}-\frac {\left (b^2 m n^2\right ) \int \left (2 d e+\frac {d^2}{x}+e^2 x\right ) \, dx}{4 e^2}\\ &=-\frac {b^2 d m n^2 x}{2 e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m \operatorname {Subst}\left (\int \left (d e \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}+e x \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx,x,d+e x\right )}{2 e^3}+\frac {(d m) \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^3}+\frac {(b m n) \operatorname {Subst}\left (\int \left (d e \left (a+b \log \left (c x^n\right )\right )-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{d-x}+e x \left (a+b \log \left (c x^n\right )\right )\right ) \, dx,x,d+e x\right )}{2 e^3}-\frac {\left (2 b^2 d m n\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^3}\\ &=-\frac {b^2 d m n^2 x}{2 e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}-\frac {(d m) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}+\frac {(d m) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2}+\frac {\left (d^2 m\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{2 e^2}-\frac {\left (d^2 m\right ) \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^2}+\frac {(b m n) \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}+\frac {(b d m n) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}-\frac {\left (2 b^2 d m n\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac {\left (b d^2 m n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{2 e^2}+\frac {\left (2 b^2 d^2 m n\right ) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {a b d m n x}{2 e}+\frac {3 b^2 d m n^2 x}{2 e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{8 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {2 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {(b m n) \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}+\frac {(b d m n) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}-\frac {(2 b d m n) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac {\left (b^2 d m n\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{2 e^2}+\frac {\left (b d^2 m n\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^2}-\frac {\left (2 b d^2 m n\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^2}-\frac {\left (b^2 d^2 m n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{2 e^2}+\frac {\left (2 b^2 d^2 m n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=-\frac {a b d m n x}{2 e}+\frac {b^2 d m n^2 x}{e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {3 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {3 b^2 d^2 m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}+\frac {\left (b^2 d m n\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac {\left (2 b^2 d m n\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}+\frac {\left (b^2 d^2 m n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac {\left (2 b^2 d^2 m n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=-\frac {a b d m n x}{2 e}+\frac {2 b^2 d m n^2 x}{e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {3 b^2 d^2 m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}-\frac {b^2 d^2 m n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e^2}\\ \end {align*}
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Mathematica [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} \log \left (f x^{m}\right ) + 2 \, a b x \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a^{2} x \log \left (f x^{m}\right ), x\right ) \]
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 \[ \int {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.08, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2} x \ln \left (f \,x^{m}\right )\, dx \]
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 \[ -\frac {1}{4} \, {\left (b^{2} {\left (m - 2 \, \log \relax (f)\right )} x^{2} - 2 \, b^{2} x^{2} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + \int \frac {2 \, {\left (b^{2} e \log \relax (c)^{2} \log \relax (f) + 2 \, a b e \log \relax (c) \log \relax (f) + a^{2} e \log \relax (f)\right )} x^{2} + 2 \, {\left (b^{2} d \log \relax (c)^{2} \log \relax (f) + 2 \, a b d \log \relax (c) \log \relax (f) + a^{2} d \log \relax (f)\right )} x + {\left ({\left (4 \, a b e \log \relax (f) + {\left (4 \, e \log \relax (c) \log \relax (f) + {\left (m n - 2 \, n \log \relax (f)\right )} e\right )} b^{2}\right )} x^{2} + 4 \, {\left (b^{2} d \log \relax (c) \log \relax (f) + a b d \log \relax (f)\right )} x - 2 \, {\left ({\left ({\left (e n - 2 \, e \log \relax (c)\right )} b^{2} - 2 \, a b e\right )} x^{2} - 2 \, {\left (b^{2} d \log \relax (c) + a b d\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) + 2 \, {\left ({\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x^{2} + {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} x\right )} \log \left (x^{m}\right )}{2 \, {\left (e x + d\right )}}\,{d x} \]
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 \[ \int x\,\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \]
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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