Optimal. Leaf size=54 \[ x \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)} \]
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Rubi [A] time = 0.0168512, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2448, 364} \[ x \log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2448
Rule 364
Rubi steps
\begin{align*} \int \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=x \log \left (c \left (d+e x^n\right )^p\right )-(e n p) \int \frac{x^n}{d+e x^n} \, dx\\ &=-\frac{e n p x^{1+n} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (1+n)}+x \log \left (c \left (d+e x^n\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0288539, size = 52, normalized size = 0.96 \[ x \left (\log \left (c \left (d+e x^n\right )^p\right )-\frac{e n p x^n \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \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] time = 0., size = 0, normalized size = 0. \begin{align*} d n p \int \frac{1}{e x^{n} + d}\,{d x} -{\left (n p - \log \left (c\right )\right )} x + x \log \left ({\left (e x^{n} + d\right )}^{p}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.95144, size = 48, normalized size = 0.89 \begin{align*} x \log{\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac{p x \Phi \left (\frac{d x^{- n} e^{i \pi }}{e}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{n \Gamma \left (1 + \frac{1}{n}\right )} \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 \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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