Optimal. Leaf size=112 \[ \frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{d^2 p x^{-2 n} (f x)^{2 n} \log \left (d+e x^n\right )}{2 e^2 f n}+\frac{d p x^{-n} (f x)^{2 n}}{2 e f n}-\frac{p (f x)^{2 n}}{4 f n} \]
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Rubi [A] time = 0.0571477, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2455, 20, 266, 43} \[ \frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{d^2 p x^{-2 n} (f x)^{2 n} \log \left (d+e x^n\right )}{2 e^2 f n}+\frac{d p x^{-n} (f x)^{2 n}}{2 e f n}-\frac{p (f x)^{2 n}}{4 f n} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 20
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (f x)^{-1+2 n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{(e p) \int \frac{x^{-1+n} (f x)^{2 n}}{d+e x^n} \, dx}{2 f}\\ &=\frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{\left (e p x^{-2 n} (f x)^{2 n}\right ) \int \frac{x^{-1+3 n}}{d+e x^n} \, dx}{2 f}\\ &=\frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{\left (e p x^{-2 n} (f x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x} \, dx,x,x^n\right )}{2 f n}\\ &=\frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{\left (e p x^{-2 n} (f x)^{2 n}\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx,x,x^n\right )}{2 f n}\\ &=-\frac{p (f x)^{2 n}}{4 f n}+\frac{d p x^{-n} (f x)^{2 n}}{2 e f n}-\frac{d^2 p x^{-2 n} (f x)^{2 n} \log \left (d+e x^n\right )}{2 e^2 f n}+\frac{(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}\\ \end{align*}
Mathematica [A] time = 0.0407131, size = 74, normalized size = 0.66 \[ -\frac{x^{-2 n} (f x)^{2 n} \left (e x^n \left (-2 e x^n \log \left (c \left (d+e x^n\right )^p\right )-2 d p+e p x^n\right )+2 d^2 p \log \left (d+e x^n\right )\right )}{4 e^2 f n} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.887, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1+2\,n}\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \, 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 = 2.09319, size = 204, normalized size = 1.82 \begin{align*} \frac{2 \, d e f^{2 \, n - 1} p x^{n} -{\left (e^{2} p - 2 \, e^{2} \log \left (c\right )\right )} f^{2 \, n - 1} x^{2 \, n} + 2 \,{\left (e^{2} f^{2 \, n - 1} p x^{2 \, n} - d^{2} f^{2 \, n - 1} p\right )} \log \left (e x^{n} + d\right )}{4 \, e^{2} 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 )^{2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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