Optimal. Leaf size=74 \[ \frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^2 n x^6-\frac {1}{32} b d e n x^8-\frac {1}{100} b e^2 n x^{10} \]
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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} \[ \frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^2 n x^6-\frac {1}{32} b d e n x^8-\frac {1}{100} b e^2 n x^{10} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 266
Rule 2334
Rubi steps
\begin {align*} \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{60} x^5 \left (10 d^2+15 d e x^2+6 e^2 x^4\right ) \, dx\\ &=\frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int x^5 \left (10 d^2+15 d e x^2+6 e^2 x^4\right ) \, dx\\ &=\frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int \left (10 d^2 x^5+15 d e x^7+6 e^2 x^9\right ) \, dx\\ &=-\frac {1}{36} b d^2 n x^6-\frac {1}{32} b d e n x^8-\frac {1}{100} b e^2 n x^{10}+\frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 84, normalized size = 1.14 \[ \frac {x^6 \left (1200 d^2 \left (a+b \log \left (c x^n\right )\right )+1800 d e x^2 \left (a+b \log \left (c x^n\right )\right )+720 e^2 x^4 \left (a+b \log \left (c x^n\right )\right )-200 b d^2 n-225 b d e n x^2-72 b e^2 n x^4\right )}{7200} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 118, normalized size = 1.59 \[ -\frac {1}{100} \, {\left (b e^{2} n - 10 \, a e^{2}\right )} x^{10} - \frac {1}{32} \, {\left (b d e n - 8 \, a d e\right )} x^{8} - \frac {1}{36} \, {\left (b d^{2} n - 6 \, a d^{2}\right )} x^{6} + \frac {1}{60} \, {\left (6 \, b e^{2} x^{10} + 15 \, b d e x^{8} + 10 \, b d^{2} x^{6}\right )} \log \relax (c) + \frac {1}{60} \, {\left (6 \, b e^{2} n x^{10} + 15 \, b d e n x^{8} + 10 \, b d^{2} n x^{6}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 123, normalized size = 1.66 \[ \frac {1}{10} \, b n x^{10} e^{2} \log \relax (x) - \frac {1}{100} \, b n x^{10} e^{2} + \frac {1}{10} \, b x^{10} e^{2} \log \relax (c) + \frac {1}{4} \, b d n x^{8} e \log \relax (x) + \frac {1}{10} \, a x^{10} e^{2} - \frac {1}{32} \, b d n x^{8} e + \frac {1}{4} \, b d x^{8} e \log \relax (c) + \frac {1}{4} \, a d x^{8} e + \frac {1}{6} \, b d^{2} n x^{6} \log \relax (x) - \frac {1}{36} \, b d^{2} n x^{6} + \frac {1}{6} \, b d^{2} x^{6} \log \relax (c) + \frac {1}{6} \, a d^{2} x^{6} \]
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 \[ -\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{20}+\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{20}+\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{20}-\frac {i \pi b \,e^{2} x^{10} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{20}-\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b d e \,x^{8} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {b \,e^{2} n \,x^{10}}{100}+\frac {b \,e^{2} x^{10} \ln \relax (c )}{10}+\frac {a \,e^{2} x^{10}}{10}-\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{12}+\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}+\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b \,d^{2} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{12}-\frac {b d e n \,x^{8}}{32}+\frac {b d e \,x^{8} \ln \relax (c )}{4}+\frac {a d e \,x^{8}}{4}-\frac {b \,d^{2} n \,x^{6}}{36}+\frac {b \,d^{2} x^{6} \ln \relax (c )}{6}+\frac {a \,d^{2} x^{6}}{6}+\frac {\left (6 e^{2} x^{4}+15 d e \,x^{2}+10 d^{2}\right ) b \,x^{6} \ln \left (x^{n}\right )}{60} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 100, normalized size = 1.35 \[ -\frac {1}{100} \, b e^{2} n x^{10} + \frac {1}{10} \, b e^{2} x^{10} \log \left (c x^{n}\right ) + \frac {1}{10} \, a e^{2} x^{10} - \frac {1}{32} \, b d e n x^{8} + \frac {1}{4} \, b d e x^{8} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d e x^{8} - \frac {1}{36} \, b d^{2} n x^{6} + \frac {1}{6} \, b d^{2} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{2} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 82, normalized size = 1.11 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^6}{6}+\frac {b\,d\,e\,x^8}{4}+\frac {b\,e^2\,x^{10}}{10}\right )+\frac {d^2\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {e^2\,x^{10}\,\left (10\,a-b\,n\right )}{100}+\frac {d\,e\,x^8\,\left (8\,a-b\,n\right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 21.66, size = 151, normalized size = 2.04 \[ \frac {a d^{2} x^{6}}{6} + \frac {a d e x^{8}}{4} + \frac {a e^{2} x^{10}}{10} + \frac {b d^{2} n x^{6} \log {\relax (x )}}{6} - \frac {b d^{2} n x^{6}}{36} + \frac {b d^{2} x^{6} \log {\relax (c )}}{6} + \frac {b d e n x^{8} \log {\relax (x )}}{4} - \frac {b d e n x^{8}}{32} + \frac {b d e x^{8} \log {\relax (c )}}{4} + \frac {b e^{2} n x^{10} \log {\relax (x )}}{10} - \frac {b e^{2} n x^{10}}{100} + \frac {b e^{2} x^{10} \log {\relax (c )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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