Optimal. Leaf size=27 \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2304} \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {1}{9} b n x^3+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.19 \[ \frac {a x^3}{3}+\frac {1}{3} b x^3 \log \left (c x^n\right )-\frac {1}{9} b n x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 30, normalized size = 1.11 \[ \frac {1}{3} \, b n x^{3} \log \relax (x) + \frac {1}{3} \, b x^{3} \log \relax (c) - \frac {1}{9} \, {\left (b n - 3 \, a\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 31, normalized size = 1.15 \[ \frac {1}{3} \, b n x^{3} \log \relax (x) - \frac {1}{9} \, b n x^{3} + \frac {1}{3} \, b x^{3} \log \relax (c) + \frac {1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 112, normalized size = 4.15 \[ \frac {b \,x^{3} \ln \left (x^{n}\right )}{3}+\frac {\left (-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b n +6 b \ln \relax (c )+6 a \right ) x^{3}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 26, normalized size = 0.96 \[ -\frac {1}{9} \, b n x^{3} + \frac {1}{3} \, b x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 25, normalized size = 0.93 \[ x^3\,\left (\frac {a}{3}-\frac {b\,n}{9}\right )+\frac {b\,x^3\,\ln \left (c\,x^n\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 36, normalized size = 1.33 \[ \frac {a x^{3}}{3} + \frac {b n x^{3} \log {\relax (x )}}{3} - \frac {b n x^{3}}{9} + \frac {b x^{3} \log {\relax (c )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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