Integrand size = 19, antiderivative size = 47 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b d n x^2-\frac {1}{16} b e n x^4+\frac {1}{4} \left (2 d x^2+e x^4\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 47, 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 = {14, 2371, 12} \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \left (2 d x^2+e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d n x^2-\frac {1}{16} b e n x^4 \]
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Rule 12
Rule 14
Rule 2371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \left (2 d x^2+e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{4} x \left (2 d+e x^2\right ) \, dx \\ & = \frac {1}{4} \left (2 d x^2+e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int x \left (2 d+e x^2\right ) \, dx \\ & = \frac {1}{4} \left (2 d x^2+e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (2 d x+e x^3\right ) \, dx \\ & = -\frac {1}{4} b d n x^2-\frac {1}{16} b e n x^4+\frac {1}{4} \left (2 d x^2+e x^4\right ) \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.47 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{2} a d x^2-\frac {1}{4} b d n x^2+\frac {1}{4} a e x^4-\frac {1}{16} b e n x^4+\frac {1}{2} b d x^2 \log \left (c x^n\right )+\frac {1}{4} b e x^4 \log \left (c x^n\right ) \]
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Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(\frac {x^{4} b e \ln \left (c \,x^{n}\right )}{4}-\frac {b e n \,x^{4}}{16}+\frac {x^{4} a e}{4}+\frac {x^{2} \ln \left (c \,x^{n}\right ) b d}{2}-\frac {b d n \,x^{2}}{4}+\frac {a d \,x^{2}}{2}\) | \(58\) |
risch | \(\text {Expression too large to display}\) | \(2346\) |
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Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, {\left (b e n - 4 \, a e\right )} x^{4} - \frac {1}{4} \, {\left (b d n - 2 \, a d\right )} x^{2} + \frac {1}{4} \, {\left (b e x^{4} + 2 \, b d x^{2}\right )} \log \left (c\right ) + \frac {1}{4} \, {\left (b e n x^{4} + 2 \, b d n x^{2}\right )} \log \left (x\right ) \]
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Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.40 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d x^{2}}{2} + \frac {a e x^{4}}{4} - \frac {b d n x^{2}}{4} + \frac {b d x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {b e n x^{4}}{16} + \frac {b e x^{4} \log {\left (c x^{n} \right )}}{4} \]
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Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b e n x^{4} + \frac {1}{4} \, b e x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a e x^{4} - \frac {1}{4} \, b d n x^{2} + \frac {1}{2} \, b d x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d x^{2} \]
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Time = 0.44 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.47 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \, b e n x^{4} \log \left (x\right ) - \frac {1}{16} \, b e n x^{4} + \frac {1}{4} \, b e x^{4} \log \left (c\right ) + \frac {1}{4} \, a e x^{4} + \frac {1}{2} \, b d n x^{2} \log \left (x\right ) - \frac {1}{4} \, b d n x^{2} + \frac {1}{2} \, b d x^{2} \log \left (c\right ) + \frac {1}{2} \, a d x^{2} \]
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Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int x \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^4}{4}+\frac {b\,d\,x^2}{2}\right )+\frac {d\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {e\,x^4\,\left (4\,a-b\,n\right )}{16} \]
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