Optimal. Leaf size=69 \[ \frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}-\frac{p (f x)^n}{f n} \]
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Rubi [A] time = 0.0428553, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2455, 20, 266, 43} \[ \frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}-\frac{p (f x)^n}{f n} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 20
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{(e p) \int \frac{x^{-1+n} (f x)^n}{d+e x^n} \, dx}{f}\\ &=\frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{\left (e p x^{-n} (f x)^n\right ) \int \frac{x^{-1+2 n}}{d+e x^n} \, dx}{f}\\ &=\frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{\left (e p x^{-n} (f x)^n\right ) \operatorname{Subst}\left (\int \frac{x}{d+e x} \, dx,x,x^n\right )}{f n}\\ &=\frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{\left (e p x^{-n} (f x)^n\right ) \operatorname{Subst}\left (\int \left (\frac{1}{e}-\frac{d}{e (d+e x)}\right ) \, dx,x,x^n\right )}{f n}\\ &=-\frac{p (f x)^n}{f n}+\frac{d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}+\frac{(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}\\ \end{align*}
Mathematica [A] time = 0.0326439, size = 48, normalized size = 0.7 \[ \frac{x^{1-n} (f x)^{n-1} \left (\frac{\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e}-p x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.869, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1+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 = 2.01271, size = 127, normalized size = 1.84 \begin{align*} -\frac{{\left (e p - e \log \left (c\right )\right )} f^{n - 1} x^{n} -{\left (e f^{n - 1} p x^{n} + d f^{n - 1} p\right )} \log \left (e x^{n} + d\right )}{e 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 )^{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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