Optimal. Leaf size=71 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {b d n}{4 x^2}-\frac {b e n x^{r-2}}{(2-r)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2334} \[ -\frac {1}{2} \left (\frac {d}{x^2}+\frac {2 e x^{r-2}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{4 x^2}-\frac {b e n x^{r-2}}{(2-r)^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2334
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac {1}{2} \left (\frac {d}{x^2}+\frac {2 e x^{-2+r}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d}{2 x^3}+\frac {e x^{-3+r}}{-2+r}\right ) \, dx\\ &=-\frac {b d n}{4 x^2}-\frac {b e n x^{-2+r}}{(2-r)^2}-\frac {1}{2} \left (\frac {d}{x^2}+\frac {2 e x^{-2+r}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 72, normalized size = 1.01 \[ -\frac {2 a (r-2) \left (d (r-2)-2 e x^r\right )+2 b (r-2) \log \left (c x^n\right ) \left (d (r-2)-2 e x^r\right )+b n \left (d (r-2)^2+4 e x^r\right )}{4 (r-2)^2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 140, normalized size = 1.97 \[ -\frac {4 \, b d n + {\left (b d n + 2 \, a d\right )} r^{2} + 8 \, a d - 4 \, {\left (b d n + 2 \, a d\right )} r + 4 \, {\left (b e n - a e r + 2 \, a e - {\left (b e r - 2 \, b e\right )} \log \relax (c) - {\left (b e n r - 2 \, b e n\right )} \log \relax (x)\right )} x^{r} + 2 \, {\left (b d r^{2} - 4 \, b d r + 4 \, b d\right )} \log \relax (c) + 2 \, {\left (b d n r^{2} - 4 \, b d n r + 4 \, b d n\right )} \log \relax (x)}{4 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 396, normalized size = 5.58 \[ -\frac {b d n r^{2} \log \relax (x)}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b n r x^{r} e \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d n r^{2}}{4 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d r^{2} \log \relax (c)}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b r x^{r} e \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, b d n r \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b n x^{r} e \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b d n r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {a d r^{2}}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b n x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {a r x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, b d r \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b x^{r} e \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b d n \log \relax (x)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d n}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, a d r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, a x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b d \log \relax (c)}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, a d}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 613, normalized size = 8.63 \[ -\frac {\left (d r -2 e \,x^{r}-2 d \right ) b \ln \left (x^{n}\right )}{2 \left (r -2\right ) x^{2}}-\frac {4 b d n +8 a e \,x^{r}-4 a e r \,x^{r}+4 b e n \,x^{r}+2 b d \,r^{2} \ln \relax (c )-8 b d r \ln \relax (c )+8 b e \,x^{r} \ln \relax (c )-8 a d r +8 a d +b d n \,r^{2}+i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a d \,r^{2}-4 b d n r +8 b d \ln \relax (c )-4 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 )-i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-2 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-4 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 )-4 b e r \,x^{r} \ln \relax (c )+2 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+4 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (r -2\right )^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.94, size = 644, normalized size = 9.07 \[ \begin {cases} - \frac {2 a d r^{2}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 a d r}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 a d}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 a e r x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 a e x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {2 b d n r^{2} \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {b d n r^{2}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 b d n r \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b d n r}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b d n \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {4 b d n}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {2 b d r^{2} \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 b d r \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b d \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b e n r x^{r} \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b e n x^{r} \log {\relax (x )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {4 b e n x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b e r x^{r} \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b e x^{r} \log {\relax (c )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} & \text {for}\: r \neq 2 \\- \frac {a d}{2 x^{2}} + a e \log {\relax (x )} + b d \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e \left (\begin {cases} - \log {\relax (c )} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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