Integrand size = 23, antiderivative size = 74 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} b d^2 n x^6-\frac {1}{32} b d e n x^8-\frac {1}{100} b e^2 n x^{10}+\frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {272, 45, 2371, 12, 14} \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^2 n x^6-\frac {1}{32} b d e n x^8-\frac {1}{100} b e^2 n x^{10} \]
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Rule 12
Rule 14
Rule 45
Rule 272
Rule 2371
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^6 \left (-200 b d^2 n-225 b d e n x^2-72 b e^2 n x^4+1200 d^2 \left (a+b \log \left (c x^n\right )\right )+1800 d e x^2 \left (a+b \log \left (c x^n\right )\right )+720 e^2 x^4 \left (a+b \log \left (c x^n\right )\right )\right )}{7200} \]
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Time = 1.37 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \(\frac {x^{10} \ln \left (c \,x^{n}\right ) b \,e^{2}}{10}-\frac {b \,e^{2} n \,x^{10}}{100}+\frac {a \,e^{2} x^{10}}{10}+\frac {x^{8} \ln \left (c \,x^{n}\right ) b d e}{4}-\frac {b d e n \,x^{8}}{32}+\frac {a d e \,x^{8}}{4}+\frac {x^{6} \ln \left (c \,x^{n}\right ) b \,d^{2}}{6}-\frac {b \,d^{2} n \,x^{6}}{36}+\frac {a \,d^{2} x^{6}}{6}\) | \(101\) |
risch | \(\frac {b \,x^{6} \left (6 e^{2} x^{4}+15 d e \,x^{2}+10 d^{2}\right ) \ln \left (x^{n}\right )}{60}-\frac {i \pi b d e \,x^{8} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b d e \,x^{8} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b \,d^{2} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b \,d^{2} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{12}+\frac {\ln \left (c \right ) b \,e^{2} x^{10}}{10}-\frac {b \,e^{2} n \,x^{10}}{100}+\frac {a \,e^{2} x^{10}}{10}-\frac {i \pi b d e \,x^{8} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}+\frac {i \pi b \,e^{2} x^{10} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{20}+\frac {i \pi b \,e^{2} x^{10} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{20}+\frac {i \pi b d e \,x^{8} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {\ln \left (c \right ) b d e \,x^{8}}{4}-\frac {b d e n \,x^{8}}{32}+\frac {a d e \,x^{8}}{4}+\frac {i \pi b \,d^{2} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b \,e^{2} x^{10} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{20}-\frac {i \pi b \,d^{2} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{12}-\frac {i \pi b \,e^{2} x^{10} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{20}+\frac {\ln \left (c \right ) b \,d^{2} x^{6}}{6}-\frac {b \,d^{2} n \,x^{6}}{36}+\frac {a \,d^{2} x^{6}}{6}\) | \(434\) |
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Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.59 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{100} \, {\left (b e^{2} n - 10 \, a e^{2}\right )} x^{10} - \frac {1}{32} \, {\left (b d e n - 8 \, a d e\right )} x^{8} - \frac {1}{36} \, {\left (b d^{2} n - 6 \, a d^{2}\right )} x^{6} + \frac {1}{60} \, {\left (6 \, b e^{2} x^{10} + 15 \, b d e x^{8} + 10 \, b d^{2} x^{6}\right )} \log \left (c\right ) + \frac {1}{60} \, {\left (6 \, b e^{2} n x^{10} + 15 \, b d e n x^{8} + 10 \, b d^{2} n x^{6}\right )} \log \left (x\right ) \]
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Time = 1.69 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.57 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{2} x^{6}}{6} + \frac {a d e x^{8}}{4} + \frac {a e^{2} x^{10}}{10} - \frac {b d^{2} n x^{6}}{36} + \frac {b d^{2} x^{6} \log {\left (c x^{n} \right )}}{6} - \frac {b d e n x^{8}}{32} + \frac {b d e x^{8} \log {\left (c x^{n} \right )}}{4} - \frac {b e^{2} n x^{10}}{100} + \frac {b e^{2} x^{10} \log {\left (c x^{n} \right )}}{10} \]
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Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{100} \, b e^{2} n x^{10} + \frac {1}{10} \, b e^{2} x^{10} \log \left (c x^{n}\right ) + \frac {1}{10} \, a e^{2} x^{10} - \frac {1}{32} \, b d e n x^{8} + \frac {1}{4} \, b d e x^{8} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d e x^{8} - \frac {1}{36} \, b d^{2} n x^{6} + \frac {1}{6} \, b d^{2} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{2} x^{6} \]
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Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{10} \, b e^{2} n x^{10} \log \left (x\right ) - \frac {1}{100} \, b e^{2} n x^{10} + \frac {1}{10} \, b e^{2} x^{10} \log \left (c\right ) + \frac {1}{10} \, a e^{2} x^{10} + \frac {1}{4} \, b d e n x^{8} \log \left (x\right ) - \frac {1}{32} \, b d e n x^{8} + \frac {1}{4} \, b d e x^{8} \log \left (c\right ) + \frac {1}{4} \, a d e x^{8} + \frac {1}{6} \, b d^{2} n x^{6} \log \left (x\right ) - \frac {1}{36} \, b d^{2} n x^{6} + \frac {1}{6} \, b d^{2} x^{6} \log \left (c\right ) + \frac {1}{6} \, a d^{2} x^{6} \]
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Time = 0.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^6}{6}+\frac {b\,d\,e\,x^8}{4}+\frac {b\,e^2\,x^{10}}{10}\right )+\frac {d^2\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {e^2\,x^{10}\,\left (10\,a-b\,n\right )}{100}+\frac {d\,e\,x^8\,\left (8\,a-b\,n\right )}{32} \]
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