Integrand size = 24, antiderivative size = 372 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 d^2 p^2 x^{1-2 n} (f x)^{-1+3 n}}{e^2 n}-\frac {d p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2}{2 e^3 n}+\frac {2 p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3}{27 e^3 n}-\frac {d^3 p^2 x^{1-3 n} (f x)^{-1+3 n} \log ^2\left (d+e x^n\right )}{3 e^3 n}-\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n} \]
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Time = 0.22 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2506, 2504, 2445, 2458, 45, 2372, 12, 14, 2338} \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 d^3 p x^{1-3 n} (f x)^{3 n-1} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}-\frac {2 d^2 p x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {d p x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {x (f x)^{3 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}-\frac {d^3 p^2 x^{1-3 n} (f x)^{3 n-1} \log ^2\left (d+e x^n\right )}{3 e^3 n}+\frac {2 d^2 p^2 x^{1-2 n} (f x)^{3 n-1}}{e^2 n}+\frac {2 p^2 x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^3}{27 e^3 n}-\frac {d p^2 x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^2}{2 e^3 n} \]
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Rule 12
Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2445
Rule 2458
Rule 2504
Rule 2506
Rubi steps \begin{align*} \text {integral}& = \left (x^{1-3 n} (f x)^{-1+3 n}\right ) \int x^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx \\ & = \frac {\left (x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int x^2 \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}-\frac {\left (2 e p x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int \frac {x^3 \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^n\right )}{3 n} \\ & = \frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}-\frac {\left (2 p x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \log \left (c x^p\right )}{x} \, dx,x,d+e x^n\right )}{3 n} \\ & = -\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {\left (2 p^2 x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e x^n\right )}{3 n} \\ & = -\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {\left (p^2 x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e x^n\right )}{9 e^3 n} \\ & = -\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {\left (p^2 x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e x^n\right )}{9 e^3 n} \\ & = \frac {2 d^2 p^2 x^{1-2 n} (f x)^{-1+3 n}}{e^2 n}-\frac {d p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2}{2 e^3 n}+\frac {2 p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3}{27 e^3 n}-\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}-\frac {\left (2 d^3 p^2 x^{1-3 n} (f x)^{-1+3 n}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x^n\right )}{3 e^3 n} \\ & = \frac {2 d^2 p^2 x^{1-2 n} (f x)^{-1+3 n}}{e^2 n}-\frac {d p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2}{2 e^3 n}+\frac {2 p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3}{27 e^3 n}-\frac {d^3 p^2 x^{1-3 n} (f x)^{-1+3 n} \log ^2\left (d+e x^n\right )}{3 e^3 n}-\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.46 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {x^{-3 n} (f x)^{3 n} \left (-18 d^3 p^2 \log ^2\left (d+e x^n\right )+6 d^3 p \log \left (d+e x^n\right ) \left (-11 p+6 \log \left (c \left (d+e x^n\right )^p\right )\right )+e x^n \left (p^2 \left (66 d^2-15 d e x^n+4 e^2 x^{2 n}\right )-6 p \left (6 d^2-3 d e x^n+2 e^2 x^{2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+18 e^2 x^{2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )\right )}{54 e^3 f n} \]
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\[\int \left (f x \right )^{-1+3 n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]
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Time = 0.36 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.72 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 \, {\left (2 \, e^{3} p^{2} - 6 \, e^{3} p \log \left (c\right ) + 9 \, e^{3} \log \left (c\right )^{2}\right )} f^{3 \, n - 1} x^{3 \, n} - 3 \, {\left (5 \, d e^{2} p^{2} - 6 \, d e^{2} p \log \left (c\right )\right )} f^{3 \, n - 1} x^{2 \, n} + 6 \, {\left (11 \, d^{2} e p^{2} - 6 \, d^{2} e p \log \left (c\right )\right )} f^{3 \, n - 1} x^{n} + 18 \, {\left (e^{3} f^{3 \, n - 1} p^{2} x^{3 \, n} + d^{3} f^{3 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 6 \, {\left (3 \, d e^{2} f^{3 \, n - 1} p^{2} x^{2 \, n} - 6 \, d^{2} e f^{3 \, n - 1} p^{2} x^{n} - 2 \, {\left (e^{3} p^{2} - 3 \, e^{3} p \log \left (c\right )\right )} f^{3 \, n - 1} x^{3 \, n} - {\left (11 \, d^{3} p^{2} - 6 \, d^{3} p \log \left (c\right )\right )} f^{3 \, n - 1}\right )} \log \left (e x^{n} + d\right )}{54 \, e^{3} n} \]
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\[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{3 n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.64 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e p {\left (\frac {6 \, d^{3} f^{3 \, n} \log \left (\frac {e x^{n} + d}{e}\right )}{e^{4} n} - \frac {2 \, e^{2} f^{3 \, n} x^{3 \, n} - 3 \, d e f^{3 \, n} x^{2 \, n} + 6 \, d^{2} f^{3 \, n} x^{n}}{e^{3} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{9 \, f} + \frac {\left (f x\right )^{3 \, n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{3 \, f n} - \frac {{\left (18 \, d^{3} f^{3 \, n} \log \left (e x^{n} + d\right )^{2} - 4 \, e^{3} f^{3 \, n} x^{3 \, n} + 15 \, d e^{2} f^{3 \, n} x^{2 \, n} - 66 \, d^{2} e f^{3 \, n} x^{n} - 6 \, {\left (6 \, f^{3 \, n} \log \left (e\right ) - 11 \, f^{3 \, n}\right )} d^{3} \log \left (e x^{n} + d\right )\right )} p^{2}}{54 \, e^{3} f n} \]
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\[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{3 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
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Timed out. \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2\,{\left (f\,x\right )}^{3\,n-1} \,d x \]
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