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 = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 364
Rule 2448
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*}
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Mathematica [A] time = 0.03, 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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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ d n p \int \frac {1}{e x^{n} + d}\,{d x} - {\left (n p - \log \relax (c)\right )} x + x \log \left ({\left (e x^{n} + d\right )}^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.43, size = 48, normalized size = 0.89 \[ x \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {p x \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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