Optimal. Leaf size=66 \[ \frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e}-\frac {2 p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e}+\frac {2 p^2 x^3}{3} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2454, 2389, 2296, 2295} \[ \frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e}-\frac {2 p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e}+\frac {2 p^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2389
Rule 2454
Rubi steps
\begin {align*} \int x^2 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^3\right )\\ &=\frac {\operatorname {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^3\right )}{3 e}\\ &=\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e}-\frac {(2 p) \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^3\right )}{3 e}\\ &=\frac {2 p^2 x^3}{3}-\frac {2 p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e}+\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 63, normalized size = 0.95 \[ \frac {1}{3} \left (\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{e}-2 p \left (\frac {\left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e}-p x^3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 96, normalized size = 1.45 \[ \frac {2 \, e p^{2} x^{3} - 2 \, e p x^{3} \log \relax (c) + e x^{3} \log \relax (c)^{2} + {\left (e p^{2} x^{3} + d p^{2}\right )} \log \left (e x^{3} + d\right )^{2} - 2 \, {\left (e p^{2} x^{3} + d p^{2} - {\left (e p x^{3} + d p\right )} \log \relax (c)\right )} \log \left (e x^{3} + d\right )}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 104, normalized size = 1.58 \[ \frac {1}{3} \, {\left ({\left (2 \, x^{3} e + {\left (x^{3} e + d\right )} \log \left (x^{3} e + d\right )^{2} - 2 \, {\left (x^{3} e + d\right )} \log \left (x^{3} e + d\right ) + 2 \, d\right )} p^{2} - 2 \, {\left (x^{3} e - {\left (x^{3} e + d\right )} \log \left (x^{3} e + d\right ) + d\right )} p \log \relax (c) + {\left (x^{3} e + d\right )} \log \relax (c)^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 1036, normalized size = 15.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 97, normalized size = 1.47 \[ \frac {1}{3} \, x^{3} \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} - \frac {2}{3} \, {\left (\frac {x^{3}}{e} - \frac {d \log \left (e x^{3} + d\right )}{e^{2}}\right )} e p \log \left ({\left (e x^{3} + d\right )}^{p} c\right ) + \frac {{\left (2 \, e x^{3} - d \log \left (e x^{3} + d\right )^{2} - 2 \, d \log \left (e x^{3} + d\right )\right )} p^{2}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 71, normalized size = 1.08 \[ \frac {2\,p^2\,x^3}{3}+{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2\,\left (\frac {d}{3\,e}+\frac {x^3}{3}\right )-\frac {2\,p\,x^3\,\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}{3}-\frac {2\,d\,p^2\,\ln \left (e\,x^3+d\right )}{3\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.88, size = 160, normalized size = 2.42 \[ \begin {cases} \frac {d p^{2} \log {\left (d + e x^{3} \right )}^{2}}{3 e} - \frac {2 d p^{2} \log {\left (d + e x^{3} \right )}}{3 e} + \frac {2 d p \log {\relax (c )} \log {\left (d + e x^{3} \right )}}{3 e} + \frac {p^{2} x^{3} \log {\left (d + e x^{3} \right )}^{2}}{3} - \frac {2 p^{2} x^{3} \log {\left (d + e x^{3} \right )}}{3} + \frac {2 p^{2} x^{3}}{3} + \frac {2 p x^{3} \log {\relax (c )} \log {\left (d + e x^{3} \right )}}{3} - \frac {2 p x^{3} \log {\relax (c )}}{3} + \frac {x^{3} \log {\relax (c )}^{2}}{3} & \text {for}\: e \neq 0 \\\frac {x^{3} \log {\left (c d^{p} \right )}^{2}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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