Optimal. Leaf size=52 \[ \frac {1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac {2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac {2}{27} n^2 p^2 x^3 \]
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Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2305, 2304, 2445} \[ \frac {1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac {2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac {2}{27} n^2 p^2 x^3 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rule 2445
Rubi steps
\begin {align*} \int x^2 \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname {Subst}\left (\int x^2 \log ^2\left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac {1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\operatorname {Subst}\left (\frac {1}{3} (2 n p) \int x^2 \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac {2}{27} n^2 p^2 x^3-\frac {2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac {1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 52, normalized size = 1.00 \[ \frac {1}{3} x^3 \log ^2\left (c \left (b x^n\right )^p\right )-\frac {2}{9} n p x^3 \log \left (c \left (b x^n\right )^p\right )+\frac {2}{27} n^2 p^2 x^3 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 113, normalized size = 2.17 \[ \frac {1}{3} \, n^{2} p^{2} x^{3} \log \relax (x)^{2} + \frac {2}{27} \, n^{2} p^{2} x^{3} - \frac {2}{9} \, n p^{2} x^{3} \log \relax (b) + \frac {1}{3} \, p^{2} x^{3} \log \relax (b)^{2} + \frac {1}{3} \, x^{3} \log \relax (c)^{2} - \frac {2}{9} \, {\left (n p x^{3} - 3 \, p x^{3} \log \relax (b)\right )} \log \relax (c) - \frac {2}{9} \, {\left (n^{2} p^{2} x^{3} - 3 \, n p^{2} x^{3} \log \relax (b) - 3 \, n p x^{3} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 115, normalized size = 2.21 \[ \frac {1}{3} \, n^{2} p^{2} x^{3} \log \relax (x)^{2} - \frac {2}{9} \, n^{2} p^{2} x^{3} \log \relax (x) + \frac {2}{3} \, n p^{2} x^{3} \log \relax (b) \log \relax (x) + \frac {2}{27} \, n^{2} p^{2} x^{3} - \frac {2}{9} \, n p^{2} x^{3} \log \relax (b) + \frac {1}{3} \, p^{2} x^{3} \log \relax (b)^{2} + \frac {2}{3} \, n p x^{3} \log \relax (c) \log \relax (x) - \frac {2}{9} \, n p x^{3} \log \relax (c) + \frac {2}{3} \, p x^{3} \log \relax (b) \log \relax (c) + \frac {1}{3} \, x^{3} \log \relax (c)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int x^{2} \ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 46, normalized size = 0.88 \[ \frac {2}{27} \, n^{2} p^{2} x^{3} - \frac {2}{9} \, n p x^{3} \log \left (\left (b x^{n}\right )^{p} c\right ) + \frac {1}{3} \, x^{3} \log \left (\left (b x^{n}\right )^{p} c\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 46, normalized size = 0.88 \[ \frac {2\,n^2\,p^2\,x^3}{27}-\frac {2\,n\,p\,x^3\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{9}+\frac {x^3\,{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.00, size = 150, normalized size = 2.88 \[ \frac {n^{2} p^{2} x^{3} \log {\relax (x )}^{2}}{3} - \frac {2 n^{2} p^{2} x^{3} \log {\relax (x )}}{9} + \frac {2 n^{2} p^{2} x^{3}}{27} + \frac {2 n p^{2} x^{3} \log {\relax (b )} \log {\relax (x )}}{3} - \frac {2 n p^{2} x^{3} \log {\relax (b )}}{9} + \frac {2 n p x^{3} \log {\relax (c )} \log {\relax (x )}}{3} - \frac {2 n p x^{3} \log {\relax (c )}}{9} + \frac {p^{2} x^{3} \log {\relax (b )}^{2}}{3} + \frac {2 p x^{3} \log {\relax (b )} \log {\relax (c )}}{3} + \frac {x^{3} \log {\relax (c )}^{2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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