Optimal. Leaf size=52 \[ \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 n^2 x^2 \]
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Rubi [A] time = 0.02, antiderivative size = 52, 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 = {2305, 2304} \[ \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 n^2 x^2 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {1}{4} b^2 n^2 x^2-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.79 \[ \frac {1}{4} x^2 \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 102, normalized size = 1.96 \[ \frac {1}{2} \, b^{2} n^{2} x^{2} \log \relax (x)^{2} + \frac {1}{2} \, b^{2} x^{2} \log \relax (c)^{2} - \frac {1}{2} \, {\left (b^{2} n - 2 \, a b\right )} x^{2} \log \relax (c) + \frac {1}{4} \, {\left (b^{2} n^{2} - 2 \, a b n + 2 \, a^{2}\right )} x^{2} + \frac {1}{2} \, {\left (2 \, b^{2} n x^{2} \log \relax (c) - {\left (b^{2} n^{2} - 2 \, a b n\right )} x^{2}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 108, normalized size = 2.08 \[ \frac {1}{2} \, b^{2} n^{2} x^{2} \log \relax (x)^{2} - \frac {1}{2} \, b^{2} n^{2} x^{2} \log \relax (x) + b^{2} n x^{2} \log \relax (c) \log \relax (x) + \frac {1}{4} \, b^{2} n^{2} x^{2} - \frac {1}{2} \, b^{2} n x^{2} \log \relax (c) + \frac {1}{2} \, b^{2} x^{2} \log \relax (c)^{2} + a b n x^{2} \log \relax (x) - \frac {1}{2} \, a b n x^{2} + a b x^{2} \log \relax (c) + \frac {1}{2} \, a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 692, normalized size = 13.31 \[ \frac {b^{2} x^{2} \ln \left (x^{n}\right )^{2}}{2}+\frac {\left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-b n +2 b \ln \relax (c )+2 a \right ) b \,x^{2} \ln \left (x^{n}\right )}{2}+\frac {\left (-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-4 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 a^{2}-2 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 b^{2} n^{2}-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+8 a b \ln \relax (c )-4 b^{2} n \ln \relax (c )+4 b^{2} \ln \relax (c )^{2}-4 a b n -\pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )-4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )\right ) x^{2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 70, normalized size = 1.35 \[ \frac {1}{2} \, b^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b n x^{2} + a b x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 60, normalized size = 1.15 \[ x^2\,\left (\frac {a^2}{2}-\frac {a\,b\,n}{2}+\frac {b^2\,n^2}{4}\right )+x^2\,\ln \left (c\,x^n\right )\,\left (a\,b-\frac {b^2\,n}{2}\right )+\frac {b^2\,x^2\,{\ln \left (c\,x^n\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.93, size = 126, normalized size = 2.42 \[ \frac {a^{2} x^{2}}{2} + a b n x^{2} \log {\relax (x )} - \frac {a b n x^{2}}{2} + a b x^{2} \log {\relax (c )} + \frac {b^{2} n^{2} x^{2} \log {\relax (x )}^{2}}{2} - \frac {b^{2} n^{2} x^{2} \log {\relax (x )}}{2} + \frac {b^{2} n^{2} x^{2}}{4} + b^{2} n x^{2} \log {\relax (c )} \log {\relax (x )} - \frac {b^{2} n x^{2} \log {\relax (c )}}{2} + \frac {b^{2} x^{2} \log {\relax (c )}^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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