Integrand size = 24, antiderivative size = 602 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=-\frac {a b d m n x}{2 e}+\frac {2 b^2 d m n^2 x}{e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {3 b^2 d^2 m n^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^2}-\frac {b^2 d^2 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e^2} \]
[Out]
Time = 0.89 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341, 2475, 45, 2458, 2393, 2354, 2438, 2395, 2421, 6724} \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {b d^2 m n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {a b d m n x}{2 e}-\frac {2 b d m n x (a-b n)}{e}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {3 b^2 d^2 m n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{2 e^2}-\frac {b^2 d^2 m n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {2 b^2 d m n^2 x}{e}-\frac {1}{8} b^2 m n^2 x^2 \]
[In]
[Out]
Rule 45
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2354
Rule 2393
Rule 2395
Rule 2421
Rule 2436
Rule 2437
Rule 2438
Rule 2448
Rule 2458
Rule 2475
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \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 \text {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) \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^3}+\frac {(b m n) \text {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 ) \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^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 \text {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) \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^3}+\frac {(b m n) \text {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 ) \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^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 \text {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) \text {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) \text {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 ) \text {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 ) \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^2}+\frac {(b m n) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \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^2}-\frac {\left (2 b d^2 m n\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^2}-\frac {\left (b^2 d^2 m n^2\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
Time = 0.17 (sec) , antiderivative size = 825, normalized size of antiderivative = 1.37 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=b^2 n^2 \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (\frac {1}{2} x^2 \log ^2(d+e x)-e \left (\frac {3 d x}{2 e^2}-\frac {x^2}{4 e}-\frac {3 d^2 \log (d+e x)}{2 e^3}-\frac {d x \log (d+e x)}{e^2}+\frac {x^2 \log (d+e x)}{2 e}+\frac {d^2 \log ^2(d+e x)}{2 e^3}\right )\right )+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (\frac {1}{2} x^2 \log (d+e x)-\frac {1}{2} e \left (-\frac {d x}{e^2}+\frac {x^2}{2 e}+\frac {d^2 \log (d+e x)}{e^3}\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )+\frac {1}{2} m x^2 \log (x) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )^2+\frac {1}{4} x^2 \left (-m+2 \left (-m \log (x)+\log \left (f x^m\right )\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )^2+b m n \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right ) \left (-\frac {1}{2} x^2 \log (d+e x)+x^2 \log (x) \log (d+e x)+\frac {1}{2} e \left (-\frac {d x}{e^2}+\frac {x^2}{2 e}+\frac {d^2 \log (d+e x)}{e^3}\right )-e \left (-\frac {d x (-1+\log (x))}{e^2}+\frac {-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)}{e}+\frac {d^2 \left (\frac {\log (x) \log \left (\frac {d+e x}{d}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{e^2}\right )\right )+\frac {1}{2} b^2 m n^2 \left (-\frac {1}{2} x^2 \log ^2(d+e x)+x^2 \log (x) \log ^2(d+e x)+e \left (\frac {3 d x}{2 e^2}-\frac {x^2}{4 e}-\frac {3 d^2 \log (d+e x)}{2 e^3}-\frac {d x \log (d+e x)}{e^2}+\frac {x^2 \log (d+e x)}{2 e}+\frac {d^2 \log ^2(d+e x)}{2 e^3}\right )-2 e \left (-\frac {d \left (2 e x-d \log (d+e x)-e x \log (d+e x)+\log (x) \left (-e x+e x \log (d+e x)+d \log \left (1+\frac {e x}{d}\right )\right )+d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )}{e^3}+\frac {-3 d e x+e^2 x^2+d^2 \log (d+e x)-e^2 x^2 \log (d+e x)+\log (x) \left (e x (2 d-e x)+2 e^2 x^2 \log (d+e x)-2 d^2 \log \left (1+\frac {e x}{d}\right )\right )-2 d^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^3}+\frac {d^2 \left (\frac {1}{2} \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log ^2(d+e x)-\log (d+e x) \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+\operatorname {PolyLog}\left (3,\frac {d+e x}{d}\right )\right )}{e^3}\right )\right ) \]
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\[\int x \ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}d x\]
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\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]
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Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Timed out} \]
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\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]
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\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]
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Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int x\,\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \]
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