Integrand size = 21, antiderivative size = 53 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {b e n x^r}{r^2}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
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Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {14, 2393, 2338, 2341} \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{r}-\frac {b e n x^r}{r^2} \]
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Rule 14
Rule 2338
Rule 2341
Rule 2393
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (a+b \log \left (c x^n\right )\right )}{x}+e x^{-1+r} \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \\ & = d \int \frac {a+b \log \left (c x^n\right )}{x} \, dx+e \int x^{-1+r} \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = -\frac {b e n x^r}{r^2}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {e (-b n+a r) x^r}{r^2}+a d \log (x)+\frac {b e x^r \log \left (c x^n\right )}{r}+\frac {b d \log ^2\left (c x^n\right )}{2 n} \]
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Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.28
method | result | size |
parallelrisch | \(\frac {2 \ln \left (x \right ) a d n \,r^{2}+2 x^{r} \ln \left (c \,x^{n}\right ) b e r n +b d \ln \left (c \,x^{n}\right )^{2} r^{2}+2 x^{r} a e n r -2 x^{r} b e \,n^{2}}{2 r^{2} n}\) | \(68\) |
risch | \(\frac {b \left (d r \ln \left (x \right )+e \,x^{r}\right ) \ln \left (x^{n}\right )}{r}-\frac {i \ln \left (x \right ) \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \ln \left (x \right ) \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \ln \left (x \right ) \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{r}}{2 r}+\frac {i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}}{2 r}+\frac {i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}}{2 r}-\frac {i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}}{2 r}-\frac {b d n \ln \left (x \right )^{2}}{2}+\ln \left (x \right ) \ln \left (c \right ) b d +\frac {\ln \left (c \right ) b e \,x^{r}}{r}+\ln \left (x \right ) a d +\frac {x^{r} a e}{r}-\frac {b e n \,x^{r}}{r^{2}}\) | \(278\) |
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Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {b d n r^{2} \log \left (x\right )^{2} + 2 \, {\left (b e n r \log \left (x\right ) + b e r \log \left (c\right ) - b e n + a e r\right )} x^{r} + 2 \, {\left (b d r^{2} \log \left (c\right ) + a d r^{2}\right )} \log \left (x\right )}{2 \, r^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (46) = 92\).
Time = 2.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.47 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \left (a + b \log {\left (c \right )}\right ) \left (d + e\right ) \log {\left (x \right )} & \text {for}\: n = 0 \wedge r = 0 \\\left (a + b \log {\left (c \right )}\right ) \left (d \log {\left (x \right )} + \frac {e x^{r}}{r}\right ) & \text {for}\: n = 0 \\\left (d + e\right ) \left (\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right ) & \text {for}\: r = 0 \\\frac {a d \log {\left (c x^{n} \right )}}{n} + \frac {a e x^{r}}{r} + \frac {b d \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {b e n x^{r}}{r^{2}} + \frac {b e x^{r} \log {\left (c x^{n} \right )}}{r} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {b e x^{r} \log \left (c x^{n}\right )}{r} + \frac {b d \log \left (c x^{n}\right )^{2}}{2 \, n} + a d \log \left (x\right ) - \frac {b e n x^{r}}{r^{2}} + \frac {a e x^{r}}{r} \]
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\[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
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