Optimal. Leaf size=52 \[ \frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{32} b^2 n^2 x^4 \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2305, 2304} \[ \frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{32} b^2 n^2 x^4 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {1}{32} b^2 n^2 x^4-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.83 \[ \frac {1}{32} x^4 \left (-4 b n \left (a+b \log \left (c x^n\right )\right )+8 \left (a+b \log \left (c x^n\right )\right )^2+b^2 n^2\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 102, normalized size = 1.96 \[ \frac {1}{4} \, b^{2} n^{2} x^{4} \log \relax (x)^{2} + \frac {1}{4} \, b^{2} x^{4} \log \relax (c)^{2} - \frac {1}{8} \, {\left (b^{2} n - 4 \, a b\right )} x^{4} \log \relax (c) + \frac {1}{32} \, {\left (b^{2} n^{2} - 4 \, a b n + 8 \, a^{2}\right )} x^{4} + \frac {1}{8} \, {\left (4 \, b^{2} n x^{4} \log \relax (c) - {\left (b^{2} n^{2} - 4 \, a b n\right )} x^{4}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 111, normalized size = 2.13 \[ \frac {1}{4} \, b^{2} n^{2} x^{4} \log \relax (x)^{2} - \frac {1}{8} \, b^{2} n^{2} x^{4} \log \relax (x) + \frac {1}{2} \, b^{2} n x^{4} \log \relax (c) \log \relax (x) + \frac {1}{32} \, b^{2} n^{2} x^{4} - \frac {1}{8} \, b^{2} n x^{4} \log \relax (c) + \frac {1}{4} \, b^{2} x^{4} \log \relax (c)^{2} + \frac {1}{2} \, a b n x^{4} \log \relax (x) - \frac {1}{8} \, a b n x^{4} + \frac {1}{2} \, a b x^{4} \log \relax (c) + \frac {1}{4} \, a^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 691, normalized size = 13.29 \[ \frac {b^{2} x^{4} \ln \left (x^{n}\right )^{2}}{4}+\frac {\left (-2 i \pi 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 \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-b n +4 b \ln \relax (c )+4 a \right ) b \,x^{4} \ln \left (x^{n}\right )}{8}+\frac {\left (-2 \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}+4 \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}+4 \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}-8 \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}-8 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-8 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+8 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}+8 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+8 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+8 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi a b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+b^{2} n^{2}-2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+4 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-2 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+4 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+16 a b \ln \relax (c )-4 b^{2} n \ln \relax (c )+8 b^{2} \ln \relax (c )^{2}-4 a b n -2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-8 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 )-8 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^{4}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 71, normalized size = 1.37 \[ \frac {1}{4} \, b^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac {1}{8} \, a b n x^{4} + \frac {1}{2} \, a b x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{32} \, {\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 61, normalized size = 1.17 \[ x^4\,\left (\frac {a^2}{4}-\frac {a\,b\,n}{8}+\frac {b^2\,n^2}{32}\right )+\frac {x^4\,\ln \left (c\,x^n\right )\,\left (a\,b-\frac {b^2\,n}{4}\right )}{2}+\frac {b^2\,x^4\,{\ln \left (c\,x^n\right )}^2}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.40, size = 131, normalized size = 2.52 \[ \frac {a^{2} x^{4}}{4} + \frac {a b n x^{4} \log {\relax (x )}}{2} - \frac {a b n x^{4}}{8} + \frac {a b x^{4} \log {\relax (c )}}{2} + \frac {b^{2} n^{2} x^{4} \log {\relax (x )}^{2}}{4} - \frac {b^{2} n^{2} x^{4} \log {\relax (x )}}{8} + \frac {b^{2} n^{2} x^{4}}{32} + \frac {b^{2} n x^{4} \log {\relax (c )} \log {\relax (x )}}{2} - \frac {b^{2} n x^{4} \log {\relax (c )}}{8} + \frac {b^{2} x^{4} \log {\relax (c )}^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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