Optimal. Leaf size=74 \[ \frac {1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d^2 n x^4-\frac {1}{18} b d e n x^6-\frac {1}{64} b e^2 n x^8 \]
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Rubi [A] time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 12, 14} \[ \frac {1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d^2 n x^4-\frac {1}{18} b d e n x^6-\frac {1}{64} b e^2 n x^8 \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 266
Rule 2334
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{24} x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \, dx\\ &=\frac {1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{24} (b n) \int x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \, dx\\ &=\frac {1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{24} (b n) \int \left (6 d^2 x^3+8 d e x^5+3 e^2 x^7\right ) \, dx\\ &=-\frac {1}{16} b d^2 n x^4-\frac {1}{18} b d e n x^6-\frac {1}{64} b e^2 n x^8+\frac {1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 87, normalized size = 1.18 \[ \frac {1}{576} x^4 \left (24 a \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+24 b \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \log \left (c x^n\right )-b n \left (36 d^2+32 d e x^2+9 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 118, normalized size = 1.59 \[ -\frac {1}{64} \, {\left (b e^{2} n - 8 \, a e^{2}\right )} x^{8} - \frac {1}{18} \, {\left (b d e n - 6 \, a d e\right )} x^{6} - \frac {1}{16} \, {\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} + \frac {1}{24} \, {\left (3 \, b e^{2} x^{8} + 8 \, b d e x^{6} + 6 \, b d^{2} x^{4}\right )} \log \relax (c) + \frac {1}{24} \, {\left (3 \, b e^{2} n x^{8} + 8 \, b d e n x^{6} + 6 \, b d^{2} n x^{4}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 123, normalized size = 1.66 \[ \frac {1}{8} \, b n x^{8} e^{2} \log \relax (x) - \frac {1}{64} \, b n x^{8} e^{2} + \frac {1}{8} \, b x^{8} e^{2} \log \relax (c) + \frac {1}{3} \, b d n x^{6} e \log \relax (x) + \frac {1}{8} \, a x^{8} e^{2} - \frac {1}{18} \, b d n x^{6} e + \frac {1}{3} \, b d x^{6} e \log \relax (c) + \frac {1}{3} \, a d x^{6} e + \frac {1}{4} \, b d^{2} n x^{4} \log \relax (x) - \frac {1}{16} \, b d^{2} n x^{4} + \frac {1}{4} \, b d^{2} x^{4} \log \relax (c) + \frac {1}{4} \, a d^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 434, normalized size = 5.86 \[ -\frac {i \pi b \,e^{2} x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{16}+\frac {i \pi b \,e^{2} x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{16}+\frac {i \pi b \,e^{2} x^{8} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{16}-\frac {i \pi b \,e^{2} x^{8} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{16}-\frac {i \pi b d e \,x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}+\frac {i \pi b d e \,x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b d e \,x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i \pi b d e \,x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {b \,e^{2} n \,x^{8}}{64}+\frac {b \,e^{2} x^{8} \ln \relax (c )}{8}+\frac {a \,e^{2} x^{8}}{8}-\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {b d e n \,x^{6}}{18}+\frac {b d e \,x^{6} \ln \relax (c )}{3}+\frac {a d e \,x^{6}}{3}-\frac {b \,d^{2} n \,x^{4}}{16}+\frac {b \,d^{2} x^{4} \ln \relax (c )}{4}+\frac {a \,d^{2} x^{4}}{4}+\frac {\left (3 e^{2} x^{4}+8 d e \,x^{2}+6 d^{2}\right ) b \,x^{4} \ln \left (x^{n}\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 100, normalized size = 1.35 \[ -\frac {1}{64} \, b e^{2} n x^{8} + \frac {1}{8} \, b e^{2} x^{8} \log \left (c x^{n}\right ) + \frac {1}{8} \, a e^{2} x^{8} - \frac {1}{18} \, b d e n x^{6} + \frac {1}{3} \, b d e x^{6} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d e x^{6} - \frac {1}{16} \, b d^{2} n x^{4} + \frac {1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 82, normalized size = 1.11 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^4}{4}+\frac {b\,d\,e\,x^6}{3}+\frac {b\,e^2\,x^8}{8}\right )+\frac {d^2\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {e^2\,x^8\,\left (8\,a-b\,n\right )}{64}+\frac {d\,e\,x^6\,\left (6\,a-b\,n\right )}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.64, size = 151, normalized size = 2.04 \[ \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} n x^{4} \log {\relax (x )}}{4} - \frac {b d^{2} n x^{4}}{16} + \frac {b d^{2} x^{4} \log {\relax (c )}}{4} + \frac {b d e n x^{6} \log {\relax (x )}}{3} - \frac {b d e n x^{6}}{18} + \frac {b d e x^{6} \log {\relax (c )}}{3} + \frac {b e^{2} n x^{8} \log {\relax (x )}}{8} - \frac {b e^{2} n x^{8}}{64} + \frac {b e^{2} x^{8} \log {\relax (c )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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