Integrand size = 19, antiderivative size = 48 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d n x^4-\frac {1}{25} b e n x^5+\frac {1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {45, 2371, 12} \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d n x^4-\frac {1}{25} b e n x^5 \]
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Rule 12
Rule 45
Rule 2371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{20} x^3 (5 d+4 e x) \, dx \\ & = \frac {1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{20} (b n) \int x^3 (5 d+4 e x) \, dx \\ & = \frac {1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{20} (b n) \int \left (5 d x^3+4 e x^4\right ) \, dx \\ & = -\frac {1}{16} b d n x^4-\frac {1}{25} b e n x^5+\frac {1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{400} x^4 \left (20 a (5 d+4 e x)-b n (25 d+16 e x)+20 b (5 d+4 e x) \log \left (c x^n\right )\right ) \]
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Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21
method | result | size |
parallelrisch | \(\frac {x^{5} \ln \left (c \,x^{n}\right ) b e}{5}-\frac {b e n \,x^{5}}{25}+\frac {a e \,x^{5}}{5}+\frac {x^{4} \ln \left (c \,x^{n}\right ) b d}{4}-\frac {b d n \,x^{4}}{16}+\frac {a d \,x^{4}}{4}\) | \(58\) |
risch | \(\frac {b \,x^{4} \left (4 e x +5 d \right ) \ln \left (x^{n}\right )}{20}-\frac {i \pi b e \,x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{10}+\frac {i \pi b e \,x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b e \,x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b e \,x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\frac {\ln \left (c \right ) b e \,x^{5}}{5}-\frac {b e n \,x^{5}}{25}+\frac {a e \,x^{5}}{5}-\frac {i \pi b d \,x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b d \,x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b d \,x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b d \,x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}+\frac {\ln \left (c \right ) b d \,x^{4}}{4}-\frac {b d n \,x^{4}}{16}+\frac {a d \,x^{4}}{4}\) | \(264\) |
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Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, {\left (b e n - 5 \, a e\right )} x^{5} - \frac {1}{16} \, {\left (b d n - 4 \, a d\right )} x^{4} + \frac {1}{20} \, {\left (4 \, b e x^{5} + 5 \, b d x^{4}\right )} \log \left (c\right ) + \frac {1}{20} \, {\left (4 \, b e n x^{5} + 5 \, b d n x^{4}\right )} \log \left (x\right ) \]
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Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d x^{4}}{4} + \frac {a e x^{5}}{5} - \frac {b d n x^{4}}{16} + \frac {b d x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {b e n x^{5}}{25} + \frac {b e x^{5} \log {\left (c x^{n} \right )}}{5} \]
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Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, b e n x^{5} + \frac {1}{5} \, b e x^{5} \log \left (c x^{n}\right ) - \frac {1}{16} \, b d n x^{4} + \frac {1}{5} \, a e x^{5} + \frac {1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d x^{4} \]
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Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{5} \, b e n x^{5} \log \left (x\right ) - \frac {1}{25} \, b e n x^{5} + \frac {1}{5} \, b e x^{5} \log \left (c\right ) + \frac {1}{4} \, b d n x^{4} \log \left (x\right ) - \frac {1}{16} \, b d n x^{4} + \frac {1}{5} \, a e x^{5} + \frac {1}{4} \, b d x^{4} \log \left (c\right ) + \frac {1}{4} \, a d x^{4} \]
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Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^5}{5}+\frac {b\,d\,x^4}{4}\right )+\frac {d\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {e\,x^5\,\left (5\,a-b\,n\right )}{25} \]
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