Integrand size = 23, antiderivative size = 127 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{-7+r}}{(7-r)^2}-\frac {b e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r} \]
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Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2372, 12, 14} \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{r-7}}{(7-r)^2}-\frac {b e^2 n x^{2 r-7}}{(7-2 r)^2} \]
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Rule 12
Rule 14
Rule 276
Rule 2372
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-(b n) \int \frac {-d^2+\frac {14 d e x^r}{-7+r}+\frac {7 e^2 x^{2 r}}{-7+2 r}}{7 x^8} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {1}{7} (b n) \int \frac {-d^2+\frac {14 d e x^r}{-7+r}+\frac {7 e^2 x^{2 r}}{-7+2 r}}{x^8} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {1}{7} (b n) \int \left (-\frac {d^2}{x^8}+\frac {14 d e x^{-8+r}}{-7+r}+\frac {7 e^2 x^{2 (-4+r)}}{-7+2 r}\right ) \, dx \\ & = -\frac {b d^2 n}{49 x^7}-\frac {2 b d e n x^{-7+r}}{(7-r)^2}-\frac {b e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {b n \left (-d^2-\frac {98 d e x^r}{(-7+r)^2}-\frac {49 e^2 x^{2 r}}{(7-2 r)^2}\right )+a \left (-7 d^2+\frac {98 d e x^r}{-7+r}+\frac {49 e^2 x^{2 r}}{-7+2 r}\right )+7 b \left (-d^2+\frac {14 d e x^r}{-7+r}+\frac {7 e^2 x^{2 r}}{-7+2 r}\right ) \log \left (c x^n\right )}{49 x^7} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(123)=246\).
Time = 3.26 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.72
method | result | size |
parallelrisch | \(-\frac {16807 b \ln \left (c \,x^{n}\right ) d^{2}+4802 b d e n \,x^{r}+637 b \,d^{2} n \,r^{2}+33614 d e \,x^{r} a -2058 b \,d^{2} n r +33614 d e \,x^{r} b \ln \left (c \,x^{n}\right )+28 a \,d^{2} r^{4}-588 a \,d^{2} r^{3}-392 a d e \,r^{3} x^{r}+2401 b \,d^{2} n +16807 a \,d^{2}+28 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-588 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}+4459 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-14406 \ln \left (c \,x^{n}\right ) b \,d^{2} r +4 b \,d^{2} n \,r^{4}-84 b \,d^{2} n \,r^{3}-2744 b d e n r \,x^{r}+4459 a \,d^{2} r^{2}-14406 a \,d^{2} r +1715 a \,e^{2} r^{2} x^{2 r}-9604 a \,e^{2} r \,x^{2 r}+2401 b \,e^{2} n \,x^{2 r}-98 a \,e^{2} r^{3} x^{2 r}+16807 e^{2} x^{2 r} b \ln \left (c \,x^{n}\right )+392 b d e n \,r^{2} x^{r}+5488 a d e \,r^{2} x^{r}-24010 a d e r \,x^{r}-686 b \,e^{2} n r \,x^{2 r}-98 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}+1715 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-9604 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +49 b \,e^{2} n \,r^{2} x^{2 r}+16807 e^{2} x^{2 r} a -392 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}+5488 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-24010 x^{r} \ln \left (c \,x^{n}\right ) b d e r}{49 x^{7} \left (-7+2 r \right )^{2} \left (-7+r \right )^{2}}\) | \(473\) |
risch | \(\text {Expression too large to display}\) | \(1930\) |
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (118) = 236\).
Time = 0.32 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.67 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {4 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{4} + 2401 \, b d^{2} n - 84 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{3} + 16807 \, a d^{2} + 637 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r^{2} - 2058 \, {\left (b d^{2} n + 7 \, a d^{2}\right )} r - 49 \, {\left (2 \, a e^{2} r^{3} - 49 \, b e^{2} n - 343 \, a e^{2} - {\left (b e^{2} n + 35 \, a e^{2}\right )} r^{2} + 14 \, {\left (b e^{2} n + 14 \, a e^{2}\right )} r + {\left (2 \, b e^{2} r^{3} - 35 \, b e^{2} r^{2} + 196 \, b e^{2} r - 343 \, b e^{2}\right )} \log \left (c\right ) + {\left (2 \, b e^{2} n r^{3} - 35 \, b e^{2} n r^{2} + 196 \, b e^{2} n r - 343 \, b e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 98 \, {\left (4 \, a d e r^{3} - 49 \, b d e n - 343 \, a d e - 4 \, {\left (b d e n + 14 \, a d e\right )} r^{2} + 7 \, {\left (4 \, b d e n + 35 \, a d e\right )} r + {\left (4 \, b d e r^{3} - 56 \, b d e r^{2} + 245 \, b d e r - 343 \, b d e\right )} \log \left (c\right ) + {\left (4 \, b d e n r^{3} - 56 \, b d e n r^{2} + 245 \, b d e n r - 343 \, b d e n\right )} \log \left (x\right )\right )} x^{r} + 7 \, {\left (4 \, b d^{2} r^{4} - 84 \, b d^{2} r^{3} + 637 \, b d^{2} r^{2} - 2058 \, b d^{2} r + 2401 \, b d^{2}\right )} \log \left (c\right ) + 7 \, {\left (4 \, b d^{2} n r^{4} - 84 \, b d^{2} n r^{3} + 637 \, b d^{2} n r^{2} - 2058 \, b d^{2} n r + 2401 \, b d^{2} n\right )} \log \left (x\right )}{49 \, {\left (4 \, r^{4} - 84 \, r^{3} + 637 \, r^{2} - 2058 \, r + 2401\right )} x^{7}} \]
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Timed out. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]
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