Integrand size = 21, antiderivative size = 48 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7+\frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2371} \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7 \]
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Rule 14
Rule 2371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {d x^4}{5}+\frac {e x^6}{7}\right ) \, dx \\ & = -\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7+\frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{5} a d x^5-\frac {1}{25} b d n x^5+\frac {1}{7} a e x^7-\frac {1}{49} b e n x^7+\frac {1}{5} b d x^5 \log \left (c x^n\right )+\frac {1}{7} b e x^7 \log \left (c x^n\right ) \]
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Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21
method | result | size |
parallelrisch | \(\frac {x^{7} b e \ln \left (c \,x^{n}\right )}{7}-\frac {b e n \,x^{7}}{49}+\frac {x^{7} a e}{7}+\frac {x^{5} b \ln \left (c \,x^{n}\right ) d}{5}-\frac {b d n \,x^{5}}{25}+\frac {x^{5} a d}{5}\) | \(58\) |
risch | \(\frac {b \,x^{5} \left (5 e \,x^{2}+7 d \right ) \ln \left (x^{n}\right )}{35}-\frac {i \pi b e \,x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{14}+\frac {i \pi b e \,x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}+\frac {i \pi b e \,x^{7} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {i \pi b e \,x^{7} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{14}+\frac {\ln \left (c \right ) b e \,x^{7}}{7}-\frac {b e n \,x^{7}}{49}+\frac {x^{7} a e}{7}-\frac {i \pi b d \,x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{10}+\frac {i \pi b d \,x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b d \,x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b d \,x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\frac {\ln \left (c \right ) b d \,x^{5}}{5}-\frac {b d n \,x^{5}}{25}+\frac {x^{5} a d}{5}\) | \(266\) |
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Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{49} \, {\left (b e n - 7 \, a e\right )} x^{7} - \frac {1}{25} \, {\left (b d n - 5 \, a d\right )} x^{5} + \frac {1}{35} \, {\left (5 \, b e x^{7} + 7 \, b d x^{5}\right )} \log \left (c\right ) + \frac {1}{35} \, {\left (5 \, b e n x^{7} + 7 \, b d n x^{5}\right )} \log \left (x\right ) \]
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Time = 0.68 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} - \frac {b d n x^{5}}{25} + \frac {b d x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {b e n x^{7}}{49} + \frac {b e x^{7} \log {\left (c x^{n} \right )}}{7} \]
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Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{49} \, b e n x^{7} + \frac {1}{7} \, b e x^{7} \log \left (c x^{n}\right ) + \frac {1}{7} \, a e x^{7} - \frac {1}{25} \, b d n x^{5} + \frac {1}{5} \, b d x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d x^{5} \]
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Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{7} \, b e n x^{7} \log \left (x\right ) - \frac {1}{49} \, b e n x^{7} + \frac {1}{7} \, b e x^{7} \log \left (c\right ) + \frac {1}{7} \, a e x^{7} + \frac {1}{5} \, b d n x^{5} \log \left (x\right ) - \frac {1}{25} \, b d n x^{5} + \frac {1}{5} \, b d x^{5} \log \left (c\right ) + \frac {1}{5} \, a d x^{5} \]
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Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^7}{7}+\frac {b\,d\,x^5}{5}\right )+\frac {d\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {e\,x^7\,\left (7\,a-b\,n\right )}{49} \]
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