Integrand size = 18, antiderivative size = 101 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-2 a b d n x+2 b^2 d n^2 x+\frac {1}{4} b^2 e n^2 x^2-2 b^2 d n x \log \left (c x^n\right )-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2 \]
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Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2367, 2333, 2332, 2342, 2341} \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=d x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-2 a b d n x-2 b^2 d n x \log \left (c x^n\right )+2 b^2 d n^2 x+\frac {1}{4} b^2 e n^2 x^2 \]
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2367
Rubi steps \begin{align*} \text {integral}& = \int \left (d \left (a+b \log \left (c x^n\right )\right )^2+e x \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx \\ & = d \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx \\ & = d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-(2 b d n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-(b e n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = -2 a b d n x+\frac {1}{4} b^2 e n^2 x^2-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-\left (2 b^2 d n\right ) \int \log \left (c x^n\right ) \, dx \\ & = -2 a b d n x+2 b^2 d n^2 x+\frac {1}{4} b^2 e n^2 x^2-2 b^2 d n x \log \left (c x^n\right )-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} x \left (b e n x \left (-2 a+b n-2 b \log \left (c x^n\right )\right )+4 d \left (a+b \log \left (c x^n\right )\right )^2+2 e x \left (a+b \log \left (c x^n\right )\right )^2-8 b d n \left (a-b n+b \log \left (c x^n\right )\right )\right ) \]
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Time = 0.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.40
method | result | size |
parallelrisch | \(\frac {b^{2} \ln \left (c \,x^{n}\right )^{2} e \,x^{2}}{2}-\frac {\ln \left (c \,x^{n}\right ) x^{2} b^{2} n e}{2}+\frac {b^{2} e \,n^{2} x^{2}}{4}+a b \ln \left (c \,x^{n}\right ) e \,x^{2}-\frac {b n a e \,x^{2}}{2}+x \,b^{2} \ln \left (c \,x^{n}\right )^{2} d -2 b^{2} d n x \ln \left (c \,x^{n}\right )+2 b^{2} d \,n^{2} x +\frac {a^{2} e \,x^{2}}{2}+2 x a b \ln \left (c \,x^{n}\right ) d -2 a b d n x +a^{2} d x\) | \(141\) |
risch | \(\text {Expression too large to display}\) | \(1545\) |
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (95) = 190\).
Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.98 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} \, {\left (b^{2} e n^{2} - 2 \, a b e n + 2 \, a^{2} e\right )} x^{2} + \frac {1}{2} \, {\left (b^{2} e x^{2} + 2 \, b^{2} d x\right )} \log \left (c\right )^{2} + \frac {1}{2} \, {\left (b^{2} e n^{2} x^{2} + 2 \, b^{2} d n^{2} x\right )} \log \left (x\right )^{2} + {\left (2 \, b^{2} d n^{2} - 2 \, a b d n + a^{2} d\right )} x - \frac {1}{2} \, {\left ({\left (b^{2} e n - 2 \, a b e\right )} x^{2} + 4 \, {\left (b^{2} d n - a b d\right )} x\right )} \log \left (c\right ) - \frac {1}{2} \, {\left ({\left (b^{2} e n^{2} - 2 \, a b e n\right )} x^{2} + 4 \, {\left (b^{2} d n^{2} - a b d n\right )} x - 2 \, {\left (b^{2} e n x^{2} + 2 \, b^{2} d n x\right )} \log \left (c\right )\right )} \log \left (x\right ) \]
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Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.61 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=a^{2} d x + \frac {a^{2} e x^{2}}{2} - 2 a b d n x + 2 a b d x \log {\left (c x^{n} \right )} - \frac {a b e n x^{2}}{2} + a b e x^{2} \log {\left (c x^{n} \right )} + 2 b^{2} d n^{2} x - 2 b^{2} d n x \log {\left (c x^{n} \right )} + b^{2} d x \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} e n^{2} x^{2}}{4} - \frac {b^{2} e n x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} e x^{2} \log {\left (c x^{n} \right )}^{2}}{2} \]
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none
Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.35 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} e x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b e n x^{2} + a b e x^{2} \log \left (c x^{n}\right ) + b^{2} d x \log \left (c x^{n}\right )^{2} - 2 \, a b d n x + \frac {1}{2} \, a^{2} e x^{2} + 2 \, a b d x \log \left (c x^{n}\right ) + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} d + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} e + a^{2} d x \]
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (95) = 190\).
Time = 0.34 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.13 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} e n^{2} x^{2} \log \left (x\right )^{2} - \frac {1}{2} \, b^{2} e n^{2} x^{2} \log \left (x\right ) + b^{2} e n x^{2} \log \left (c\right ) \log \left (x\right ) + b^{2} d n^{2} x \log \left (x\right )^{2} + \frac {1}{4} \, b^{2} e n^{2} x^{2} - \frac {1}{2} \, b^{2} e n x^{2} \log \left (c\right ) + \frac {1}{2} \, b^{2} e x^{2} \log \left (c\right )^{2} - 2 \, b^{2} d n^{2} x \log \left (x\right ) + a b e n x^{2} \log \left (x\right ) + 2 \, b^{2} d n x \log \left (c\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} x - \frac {1}{2} \, a b e n x^{2} - 2 \, b^{2} d n x \log \left (c\right ) + a b e x^{2} \log \left (c\right ) + b^{2} d x \log \left (c\right )^{2} + 2 \, a b d n x \log \left (x\right ) - 2 \, a b d n x + \frac {1}{2} \, a^{2} e x^{2} + 2 \, a b d x \log \left (c\right ) + a^{2} d x \]
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Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,\left (2\,a-b\,n\right )\,x^2}{2}+2\,b\,d\,\left (a-b\,n\right )\,x\right )+{\ln \left (c\,x^n\right )}^2\,\left (\frac {e\,b^2\,x^2}{2}+d\,b^2\,x\right )+\frac {e\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+d\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right ) \]
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