Optimal. Leaf size=65 \[ -\frac {e n p x^{3+n} \, _2F_1\left (1,\frac {3+n}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{3 d (3+n)}+\frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right ) \]
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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2505, 371}
\begin {gather*} \frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^{n+3} \, _2F_1\left (1,\frac {n+3}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{3 d (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2505
Rubi steps
\begin {align*} \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac {1}{3} (e n p) \int \frac {x^{2+n}}{d+e x^n} \, dx\\ &=-\frac {e n p x^{3+n} \, _2F_1\left (1,\frac {3+n}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{3 d (3+n)}+\frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 61, normalized size = 0.94 \begin {gather*} \frac {1}{3} x^3 \left (-\frac {e n p x^n \, _2F_1\left (1,\frac {3+n}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{d (3+n)}+\log \left (c \left (d+e x^n\right )^p\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{2} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 7.90, size = 104, normalized size = 1.60 \begin {gather*} \frac {x^{3} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{3} - \frac {e p x^{3} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{3 d \Gamma \left (2 + \frac {3}{n}\right )} - \frac {e p x^{3} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{d n \Gamma \left (2 + \frac {3}{n}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,\ln \left (c\,{\left (d+e\,x^n\right )}^p\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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