Optimal. Leaf size=65 \[ \frac {1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^{n+2} \, _2F_1\left (1,\frac {n+2}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (n+2)} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2455, 364} \[ \frac {1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^{n+2} \, _2F_1\left (1,\frac {n+2}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (n+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2455
Rubi steps
\begin {align*} \int x \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac {1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )-\frac {1}{2} (e n p) \int \frac {x^{1+n}}{d+e x^n} \, dx\\ &=-\frac {e n p x^{2+n} \, _2F_1\left (1,\frac {2+n}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (2+n)}+\frac {1}{2} x^2 \log \left (c \left (d+e x^n\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.94 \[ \frac {1}{2} x^2 \left (\log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^n \, _2F_1\left (1,\frac {n+2}{n};2+\frac {2}{n};-\frac {e x^n}{d}\right )}{d (n+2)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \]
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 \[ \int x \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.29, size = 0, normalized size = 0.00 \[ \int x \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )\, dx \]
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 \[ d n p \int \frac {x}{2 \, {\left (e x^{n} + d\right )}}\,{d x} - \frac {1}{4} \, {\left (n p - 2 \, \log \relax (c)\right )} x^{2} + \frac {1}{2} \, x^{2} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) \]
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 \[ \int x\,\ln \left (c\,{\left (d+e\,x^n\right )}^p\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.63, size = 104, normalized size = 1.60 \[ \frac {x^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2} - \frac {e p x^{2} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{2 d \Gamma \left (2 + \frac {2}{n}\right )} - \frac {e p x^{2} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{d n \Gamma \left (2 + \frac {2}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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