Optimal. Leaf size=74 \[ \frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d^2 n x^4-\frac {2}{25} b d e n x^5-\frac {1}{36} b e^2 n x^6 \]
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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 = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {43, 2334, 12, 14} \[ \frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d^2 n x^4-\frac {2}{25} b d e n x^5-\frac {1}{36} b e^2 n x^6 \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 2334
Rubi steps
\begin {align*} \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{60} x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right ) \, dx\\ &=\frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right ) \, dx\\ &=\frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int \left (15 d^2 x^3+24 d e x^4+10 e^2 x^5\right ) \, dx\\ &=-\frac {1}{16} b d^2 n x^4-\frac {2}{25} b d e n x^5-\frac {1}{36} b e^2 n x^6+\frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 1.09 \[ \frac {x^4 \left (60 a \left (15 d^2+24 d e x+10 e^2 x^2\right )+60 b \left (15 d^2+24 d e x+10 e^2 x^2\right ) \log \left (c x^n\right )-b n \left (225 d^2+288 d e x+100 e^2 x^2\right )\right )}{3600} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 118, normalized size = 1.59 \[ -\frac {1}{36} \, {\left (b e^{2} n - 6 \, a e^{2}\right )} x^{6} - \frac {2}{25} \, {\left (b d e n - 5 \, a d e\right )} x^{5} - \frac {1}{16} \, {\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} + \frac {1}{60} \, {\left (10 \, b e^{2} x^{6} + 24 \, b d e x^{5} + 15 \, b d^{2} x^{4}\right )} \log \relax (c) + \frac {1}{60} \, {\left (10 \, b e^{2} n x^{6} + 24 \, b d e n x^{5} + 15 \, b d^{2} n x^{4}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 123, normalized size = 1.66 \[ \frac {1}{6} \, b n x^{6} e^{2} \log \relax (x) + \frac {2}{5} \, b d n x^{5} e \log \relax (x) - \frac {1}{36} \, b n x^{6} e^{2} - \frac {2}{25} \, b d n x^{5} e + \frac {1}{6} \, b x^{6} e^{2} \log \relax (c) + \frac {2}{5} \, b d x^{5} e \log \relax (c) + \frac {1}{4} \, b d^{2} n x^{4} \log \relax (x) - \frac {1}{16} \, b d^{2} n x^{4} + \frac {1}{6} \, a x^{6} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{4} \, b d^{2} x^{4} \log \relax (c) + \frac {1}{4} \, a d^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 432, normalized size = 5.84 \[ -\frac {i \pi b \,e^{2} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{12}+\frac {i \pi b \,e^{2} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}+\frac {i \pi b \,e^{2} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b \,e^{2} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{12}-\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{5}+\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{5}+\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{5}-\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{5}-\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b \,d^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {b \,e^{2} n \,x^{6}}{36}+\frac {b \,e^{2} x^{6} \ln \relax (c )}{6}+\frac {a \,e^{2} x^{6}}{6}-\frac {2 b d e n \,x^{5}}{25}+\frac {2 b d e \,x^{5} \ln \relax (c )}{5}+\frac {2 a d e \,x^{5}}{5}-\frac {b \,d^{2} n \,x^{4}}{16}+\frac {b \,d^{2} x^{4} \ln \relax (c )}{4}+\frac {a \,d^{2} x^{4}}{4}+\frac {\left (10 e^{2} x^{2}+24 d e x +15 d^{2}\right ) b \,x^{4} \ln \left (x^{n}\right )}{60} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 100, normalized size = 1.35 \[ -\frac {1}{36} \, b e^{2} n x^{6} + \frac {1}{6} \, b e^{2} x^{6} \log \left (c x^{n}\right ) - \frac {2}{25} \, b d e n x^{5} + \frac {1}{6} \, a e^{2} x^{6} + \frac {2}{5} \, b d e x^{5} \log \left (c x^{n}\right ) - \frac {1}{16} \, b d^{2} n x^{4} + \frac {2}{5} \, a d e x^{5} + \frac {1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 82, normalized size = 1.11 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^4}{4}+\frac {2\,b\,d\,e\,x^5}{5}+\frac {b\,e^2\,x^6}{6}\right )+\frac {d^2\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {e^2\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {2\,d\,e\,x^5\,\left (5\,a-b\,n\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.92, size = 158, normalized size = 2.14 \[ \frac {a d^{2} x^{4}}{4} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} n x^{4} \log {\relax (x )}}{4} - \frac {b d^{2} n x^{4}}{16} + \frac {b d^{2} x^{4} \log {\relax (c )}}{4} + \frac {2 b d e n x^{5} \log {\relax (x )}}{5} - \frac {2 b d e n x^{5}}{25} + \frac {2 b d e x^{5} \log {\relax (c )}}{5} + \frac {b e^{2} n x^{6} \log {\relax (x )}}{6} - \frac {b e^{2} n x^{6}}{36} + \frac {b e^{2} x^{6} \log {\relax (c )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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