Optimal. Leaf size=53 \[ \frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{r}-\frac {b e n x^r}{r^2} \]
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Rubi [A] time = 0.09, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {14, 2351, 2301, 2304} \[ \frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{r}-\frac {b e n x^r}{r^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2301
Rule 2304
Rule 2351
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\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*}
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Mathematica [A] time = 0.09, size = 54, normalized size = 1.02 \[ \frac {e x^r (a r-b n)}{r^2}+a d \log (x)+\frac {b d \log ^2\left (c x^n\right )}{2 n}+\frac {b e x^r \log \left (c x^n\right )}{r} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 64, normalized size = 1.21 \[ \frac {b d n r^{2} \log \relax (x)^{2} + 2 \, {\left (b e n r \log \relax (x) + b e r \log \relax (c) - b e n + a e r\right )} x^{r} + 2 \, {\left (b d r^{2} \log \relax (c) + a d r^{2}\right )} \log \relax (x)}{2 \, r^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 69, normalized size = 1.30 \[ \frac {1}{2} \, b d n \log \relax (x)^{2} + \frac {b n x^{r} e \log \relax (x)}{r} + b d \log \relax (c) \log \relax (x) + \frac {b x^{r} e \log \relax (c)}{r} + a d \log \relax (x) - \frac {b n x^{r} e}{r^{2}} + \frac {a x^{r} e}{r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 278, normalized size = 5.25 \[ -\frac {i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )}{2}+\frac {i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2}-\frac {b d n \ln \relax (x )^{2}}{2}-\frac {i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 r}+\frac {i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 r}+\frac {i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 r}-\frac {i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 r}+b d \ln \relax (c ) \ln \relax (x )+a d \ln \relax (x )+\frac {b e \,x^{r} \ln \relax (c )}{r}+\frac {a e \,x^{r}}{r}-\frac {b e n \,x^{r}}{r^{2}}+\frac {\left (d r \ln \relax (x )+e \,x^{r}\right ) b \ln \left (x^{n}\right )}{r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 56, normalized size = 1.06 \[ \frac {b e x^{r} \log \left (c x^{n}\right )}{r} + \frac {b d \log \left (c x^{n}\right )^{2}}{2 \, n} + a d \log \relax (x) - \frac {b e n x^{r}}{r^{2}} + \frac {a e x^{r}}{r} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.86, size = 112, normalized size = 2.11 \[ \begin {cases} a d \log {\relax (x )} + \frac {a e x^{r}}{r} + \frac {b d n \log {\relax (x )}^{2}}{2} + b d \log {\relax (c )} \log {\relax (x )} + \frac {b e n x^{r} \log {\relax (x )}}{r} - \frac {b e n x^{r}}{r^{2}} + \frac {b e x^{r} \log {\relax (c )}}{r} & \text {for}\: r \neq 0 \\\left (d + e\right ) \left (\begin {cases} a \log {\relax (x )} & \text {for}\: b = 0 \\- \left (- a - b \log {\relax (c )}\right ) \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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