Integrand size = 19, antiderivative size = 109 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} b^2 d n^2 x^2+\frac {2}{27} b^2 e n^2 x^3-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2 \]
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Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2395, 2342, 2341} \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d n^2 x^2+\frac {2}{27} b^2 e n^2 x^3 \]
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Rule 2341
Rule 2342
Rule 2395
Rubi steps \begin{align*} \text {integral}& = \int \left (d x \left (a+b \log \left (c x^n\right )\right )^2+e x^2 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx \\ & = d \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx \\ & = \frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2-(b d n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{3} (2 b e n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \frac {1}{4} b^2 d n^2 x^2+\frac {2}{27} b^2 e n^2 x^3-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{108} x^2 \left (8 b e n x \left (-3 a+b n-3 b \log \left (c x^n\right )\right )+27 b d n \left (-2 a+b n-2 b \log \left (c x^n\right )\right )+54 d \left (a+b \log \left (c x^n\right )\right )^2+36 e x \left (a+b \log \left (c x^n\right )\right )^2\right ) \]
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Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(\frac {x^{3} b^{2} \ln \left (c \,x^{n}\right )^{2} e}{3}-\frac {2 \ln \left (c \,x^{n}\right ) x^{3} b^{2} n e}{9}+\frac {2 b^{2} e \,n^{2} x^{3}}{27}+\frac {2 x^{3} a b \ln \left (c \,x^{n}\right ) e}{3}-\frac {2 b n \,x^{3} a e}{9}+\frac {x^{2} b^{2} \ln \left (c \,x^{n}\right )^{2} d}{2}-\frac {\ln \left (c \,x^{n}\right ) x^{2} b^{2} n d}{2}+\frac {b^{2} d \,n^{2} x^{2}}{4}+\frac {x^{3} a^{2} e}{3}+x^{2} a b \ln \left (c \,x^{n}\right ) d -\frac {b n a d \,x^{2}}{2}+\frac {x^{2} a^{2} d}{2}\) | \(155\) |
risch | \(\text {Expression too large to display}\) | \(1621\) |
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (97) = 194\).
Time = 0.27 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.01 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{27} \, {\left (2 \, b^{2} e n^{2} - 6 \, a b e n + 9 \, a^{2} e\right )} x^{3} + \frac {1}{4} \, {\left (b^{2} d n^{2} - 2 \, a b d n + 2 \, a^{2} d\right )} x^{2} + \frac {1}{6} \, {\left (2 \, b^{2} e x^{3} + 3 \, b^{2} d x^{2}\right )} \log \left (c\right )^{2} + \frac {1}{6} \, {\left (2 \, b^{2} e n^{2} x^{3} + 3 \, b^{2} d n^{2} x^{2}\right )} \log \left (x\right )^{2} - \frac {1}{18} \, {\left (4 \, {\left (b^{2} e n - 3 \, a b e\right )} x^{3} + 9 \, {\left (b^{2} d n - 2 \, a b d\right )} x^{2}\right )} \log \left (c\right ) - \frac {1}{18} \, {\left (4 \, {\left (b^{2} e n^{2} - 3 \, a b e n\right )} x^{3} + 9 \, {\left (b^{2} d n^{2} - 2 \, a b d n\right )} x^{2} - 6 \, {\left (2 \, b^{2} e n x^{3} + 3 \, b^{2} d n x^{2}\right )} \log \left (c\right )\right )} \log \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} - \frac {a b d n x^{2}}{2} + a b d x^{2} \log {\left (c x^{n} \right )} - \frac {2 a b e n x^{3}}{9} + \frac {2 a b e x^{3} \log {\left (c x^{n} \right )}}{3} + \frac {b^{2} d n^{2} x^{2}}{4} - \frac {b^{2} d n x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} d x^{2} \log {\left (c x^{n} \right )}^{2}}{2} + \frac {2 b^{2} e n^{2} x^{3}}{27} - \frac {2 b^{2} e n x^{3} \log {\left (c x^{n} \right )}}{9} + \frac {b^{2} e x^{3} \log {\left (c x^{n} \right )}^{2}}{3} \]
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Time = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.38 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b e n x^{3} + \frac {2}{3} \, a b e x^{3} \log \left (c x^{n}\right ) + \frac {1}{2} \, b^{2} d x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b d n x^{2} + \frac {1}{3} \, a^{2} e x^{3} + a b d x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d + \frac {2}{27} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} e \]
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (97) = 194\).
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.18 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} e n^{2} x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, b^{2} e n^{2} x^{3} \log \left (x\right ) + \frac {2}{3} \, b^{2} e n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, b^{2} d n^{2} x^{2} \log \left (x\right )^{2} + \frac {2}{27} \, b^{2} e n^{2} x^{3} - \frac {2}{9} \, b^{2} e n x^{3} \log \left (c\right ) + \frac {1}{3} \, b^{2} e x^{3} \log \left (c\right )^{2} - \frac {1}{2} \, b^{2} d n^{2} x^{2} \log \left (x\right ) + \frac {2}{3} \, a b e n x^{3} \log \left (x\right ) + b^{2} d n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac {1}{4} \, b^{2} d n^{2} x^{2} - \frac {2}{9} \, a b e n x^{3} - \frac {1}{2} \, b^{2} d n x^{2} \log \left (c\right ) + \frac {2}{3} \, a b e x^{3} \log \left (c\right ) + \frac {1}{2} \, b^{2} d x^{2} \log \left (c\right )^{2} + a b d n x^{2} \log \left (x\right ) - \frac {1}{2} \, a b d n x^{2} + \frac {1}{3} \, a^{2} e x^{3} + a b d x^{2} \log \left (c\right ) + \frac {1}{2} \, a^{2} d x^{2} \]
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Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06 \[ \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx={\ln \left (c\,x^n\right )}^2\,\left (\frac {e\,b^2\,x^3}{3}+\frac {d\,b^2\,x^2}{2}\right )+\ln \left (c\,x^n\right )\,\left (\frac {2\,b\,e\,\left (3\,a-b\,n\right )\,x^3}{9}+\frac {b\,d\,\left (2\,a-b\,n\right )\,x^2}{2}\right )+\frac {d\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+\frac {e\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{27} \]
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