Integrand size = 14, antiderivative size = 27 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b n x^4+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2341} \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{16} b n x^4+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a x^4}{4}-\frac {1}{16} b n x^4+\frac {1}{4} b x^4 \log \left (c x^n\right ) \]
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Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {x^{4} \ln \left (c \,x^{n}\right ) b}{4}-\frac {b n \,x^{4}}{16}+\frac {a \,x^{4}}{4}\) | \(27\) |
risch | \(\frac {b \,x^{4} \ln \left (x^{n}\right )}{4}+\frac {x^{4} \left (-2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 b \ln \left (c \right )-b n +4 a \right )}{16}\) | \(112\) |
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none
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \, b n x^{4} \log \left (x\right ) + \frac {1}{4} \, b x^{4} \log \left (c\right ) - \frac {1}{16} \, {\left (b n - 4 \, a\right )} x^{4} \]
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Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a x^{4}}{4} - \frac {b n x^{4}}{16} + \frac {b x^{4} \log {\left (c x^{n} \right )}}{4} \]
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none
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b n x^{4} + \frac {1}{4} \, b x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a x^{4} \]
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none
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \, b n x^{4} \log \left (x\right ) - \frac {1}{16} \, b n x^{4} + \frac {1}{4} \, b x^{4} \log \left (c\right ) + \frac {1}{4} \, a x^{4} \]
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=x^4\,\left (\frac {a}{4}-\frac {b\,n}{16}\right )+\frac {b\,x^4\,\ln \left (c\,x^n\right )}{4} \]
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