Optimal. Leaf size=372 \[ -\frac{2 d^2 p x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac{2 d^3 p x^{1-3 n} (f x)^{3 n-1} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}-\frac{2 p x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac{d p x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac{x (f x)^{3 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac{2 d^2 p^2 x^{1-2 n} (f x)^{3 n-1}}{e^2 n}-\frac{d^3 p^2 x^{1-3 n} (f x)^{3 n-1} \log ^2\left (d+e x^n\right )}{3 e^3 n}+\frac{2 p^2 x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^3}{27 e^3 n}-\frac{d p^2 x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^2}{2 e^3 n} \]
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Rubi [A] time = 0.320828, antiderivative size = 278, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2456, 2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{p x^{1-3 n} (f x)^{3 n-1} \left (\frac{18 d^2 \left (d+e x^n\right )}{e^3}-\frac{6 d^3 \log \left (d+e x^n\right )}{e^3}-\frac{9 d \left (d+e x^n\right )^2}{e^3}+\frac{2 \left (d+e x^n\right )^3}{e^3}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{9 n}+\frac{x (f x)^{3 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac{2 d^2 p^2 x^{1-2 n} (f x)^{3 n-1}}{e^2 n}-\frac{d^3 p^2 x^{1-3 n} (f x)^{3 n-1} \log ^2\left (d+e x^n\right )}{3 e^3 n}+\frac{2 p^2 x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^3}{27 e^3 n}-\frac{d p^2 x^{1-3 n} (f x)^{3 n-1} \left (d+e x^n\right )^2}{2 e^3 n} \]
Antiderivative was successfully verified.
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Rule 2456
Rule 2454
Rule 2398
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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{p x^{1-3 n} (f x)^{-1+3 n} \left (\frac{18 d^2 \left (d+e x^n\right )}{e^3}-\frac{9 d \left (d+e x^n\right )^2}{e^3}+\frac{2 \left (d+e x^n\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^n\right )}{e^3}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{9 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 ) \operatorname{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{p x^{1-3 n} (f x)^{-1+3 n} \left (\frac{18 d^2 \left (d+e x^n\right )}{e^3}-\frac{9 d \left (d+e x^n\right )^2}{e^3}+\frac{2 \left (d+e x^n\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^n\right )}{e^3}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{9 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 ) \operatorname{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{p x^{1-3 n} (f x)^{-1+3 n} \left (\frac{18 d^2 \left (d+e x^n\right )}{e^3}-\frac{9 d \left (d+e x^n\right )^2}{e^3}+\frac{2 \left (d+e x^n\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^n\right )}{e^3}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{9 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 ) \operatorname{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{p x^{1-3 n} (f x)^{-1+3 n} \left (\frac{18 d^2 \left (d+e x^n\right )}{e^3}-\frac{9 d \left (d+e x^n\right )^2}{e^3}+\frac{2 \left (d+e x^n\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^n\right )}{e^3}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{9 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 ) \operatorname{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{p x^{1-3 n} (f x)^{-1+3 n} \left (\frac{18 d^2 \left (d+e x^n\right )}{e^3}-\frac{9 d \left (d+e x^n\right )^2}{e^3}+\frac{2 \left (d+e x^n\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^n\right )}{e^3}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{9 n}+\frac{x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n}\\ \end{align*}
Mathematica [A] time = 0.153335, size = 171, normalized size = 0.46 \[ \frac{x^{-3 n} (f x)^{3 n} \left (e x^n \left (-6 p \left (6 d^2-3 d e x^n+2 e^2 x^{2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+18 e^2 x^{2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )+p^2 \left (66 d^2-15 d e x^n+4 e^2 x^{2 n}\right )\right )+6 d^3 p \log \left (d+e x^n\right ) \left (6 \log \left (c \left (d+e x^n\right )^p\right )-11 p\right )-18 d^3 p^2 \log ^2\left (d+e x^n\right )\right )}{54 e^3 f n} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.118, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1+3\,n} \left ( \ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12607, size = 603, normalized size = 1.62 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{3 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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