Optimal. Leaf size=27 \[ \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \]
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Rubi [A] time = 0.03, antiderivative size = 27, 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 = {2304, 2445} \[ \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2445
Rubi steps
\begin {align*} \int x^2 \log \left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname {Subst}\left (\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 {1}{9} n p x^3+\frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ \frac {1}{3} x^3 \log \left (c \left (b x^n\right )^p\right )-\frac {1}{9} n p x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 32, normalized size = 1.19 \[ \frac {1}{3} \, n p x^{3} \log \relax (x) - \frac {1}{9} \, n p x^{3} + \frac {1}{3} \, p x^{3} \log \relax (b) + \frac {1}{3} \, x^{3} \log \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 32, normalized size = 1.19 \[ \frac {1}{3} \, n p x^{3} \log \relax (x) - \frac {1}{9} \, n p x^{3} + \frac {1}{3} \, p x^{3} \log \relax (b) + \frac {1}{3} \, x^{3} \log \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int x^{2} \ln \left (c \left (b \,x^{n}\right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 23, normalized size = 0.85 \[ -\frac {1}{9} \, n p x^{3} + \frac {1}{3} \, x^{3} \log \left (\left (b x^{n}\right )^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.84, size = 23, normalized size = 0.85 \[ \frac {x^3\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{3}-\frac {n\,p\,x^3}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.21, size = 37, normalized size = 1.37 \[ \frac {n p x^{3} \log {\relax (x )}}{3} - \frac {n p x^{3}}{9} + \frac {p x^{3} \log {\relax (b )}}{3} + \frac {x^{3} \log {\relax (c )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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