Optimal. Leaf size=59 \[ \frac {1}{3} \left (d x^3+\frac {3 e x^{r+3}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d n x^3-\frac {b e n x^{r+3}}{(r+3)^2} \]
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Rubi [A] time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {14, 2334, 12} \[ \frac {1}{3} \left (d x^3+\frac {3 e x^{r+3}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d n x^3-\frac {b e n x^{r+3}}{(r+3)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2334
Rubi steps
\begin {align*} \int x^2 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{3} x^2 \left (d+\frac {3 e x^r}{3+r}\right ) \, dx\\ &=\frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int x^2 \left (d+\frac {3 e x^r}{3+r}\right ) \, dx\\ &=\frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (d x^2+\frac {3 e x^{2+r}}{3+r}\right ) \, dx\\ &=-\frac {1}{9} b d n x^3-\frac {b e n x^{3+r}}{(3+r)^2}+\frac {1}{3} \left (d x^3+\frac {3 e x^{3+r}}{3+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 73, normalized size = 1.24 \[ \frac {x^3 \left (3 a (r+3) \left (d (r+3)+3 e x^r\right )+3 b (r+3) \log \left (c x^n\right ) \left (d (r+3)+3 e x^r\right )-b n \left (d (r+3)^2+9 e x^r\right )\right )}{9 (r+3)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 159, normalized size = 2.69 \[ \frac {3 \, {\left (b d r^{2} + 6 \, b d r + 9 \, b d\right )} x^{3} \log \relax (c) + 3 \, {\left (b d n r^{2} + 6 \, b d n r + 9 \, b d n\right )} x^{3} \log \relax (x) - {\left (9 \, b d n + {\left (b d n - 3 \, a d\right )} r^{2} - 27 \, a d + 6 \, {\left (b d n - 3 \, a d\right )} r\right )} x^{3} + 9 \, {\left ({\left (b e r + 3 \, b e\right )} x^{3} \log \relax (c) + {\left (b e n r + 3 \, b e n\right )} x^{3} \log \relax (x) - {\left (b e n - a e r - 3 \, a e\right )} x^{3}\right )} x^{r}}{9 \, {\left (r^{2} + 6 \, r + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 137, normalized size = 2.32 \[ \frac {b n r x^{3} x^{r} e \log \relax (x)}{r^{2} + 6 \, r + 9} + \frac {1}{3} \, b d n x^{3} \log \relax (x) + \frac {3 \, b n x^{3} x^{r} e \log \relax (x)}{r^{2} + 6 \, r + 9} - \frac {1}{9} \, b d n x^{3} - \frac {b n x^{3} x^{r} e}{r^{2} + 6 \, r + 9} + \frac {1}{3} \, b d x^{3} \log \relax (c) + \frac {b x^{3} x^{r} e \log \relax (c)}{r + 3} + \frac {1}{3} \, a d x^{3} + \frac {a x^{3} x^{r} e}{r + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 614, normalized size = 10.41 \[ \frac {\left (d r +3 e \,x^{r}+3 d \right ) b \,x^{3} \ln \left (x^{n}\right )}{3 r +9}-\frac {\left (18 b d n -54 a e \,x^{r}-18 a e r \,x^{r}+18 b e n \,x^{r}-6 b d \,r^{2} \ln \relax (c )-36 b d r \ln \relax (c )-54 b e \,x^{r} \ln \relax (c )-36 a d r -54 a d +2 b d n \,r^{2}-3 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-6 a d \,r^{2}+12 b d n r -54 b d \ln \relax (c )+27 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 )+3 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 )-9 i \pi b e r \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e r \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+18 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 )+27 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 )-18 b e r \,x^{r} \ln \relax (c )+9 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 )-27 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+27 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+3 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b d r \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+27 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-18 i \pi b d r \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d r \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-27 i \pi b e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-27 i \pi b e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-27 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}\right ) x^{3}}{18 \left (r +3\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 76, normalized size = 1.29 \[ -\frac {1}{9} \, b d n x^{3} + \frac {1}{3} \, b d x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d x^{3} + \frac {b e x^{r + 3} \log \left (c x^{n}\right )}{r + 3} - \frac {b e n x^{r + 3}}{{\left (r + 3\right )}^{2}} + \frac {a e x^{r + 3}}{r + 3} \]
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 \[ \int x^2\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.46, size = 525, normalized size = 8.90 \[ \begin {cases} \frac {3 a d r^{2} x^{3}}{9 r^{2} + 54 r + 81} + \frac {18 a d r x^{3}}{9 r^{2} + 54 r + 81} + \frac {27 a d x^{3}}{9 r^{2} + 54 r + 81} + \frac {9 a e r x^{3} x^{r}}{9 r^{2} + 54 r + 81} + \frac {27 a e x^{3} x^{r}}{9 r^{2} + 54 r + 81} + \frac {3 b d n r^{2} x^{3} \log {\relax (x )}}{9 r^{2} + 54 r + 81} - \frac {b d n r^{2} x^{3}}{9 r^{2} + 54 r + 81} + \frac {18 b d n r x^{3} \log {\relax (x )}}{9 r^{2} + 54 r + 81} - \frac {6 b d n r x^{3}}{9 r^{2} + 54 r + 81} + \frac {27 b d n x^{3} \log {\relax (x )}}{9 r^{2} + 54 r + 81} - \frac {9 b d n x^{3}}{9 r^{2} + 54 r + 81} + \frac {3 b d r^{2} x^{3} \log {\relax (c )}}{9 r^{2} + 54 r + 81} + \frac {18 b d r x^{3} \log {\relax (c )}}{9 r^{2} + 54 r + 81} + \frac {27 b d x^{3} \log {\relax (c )}}{9 r^{2} + 54 r + 81} + \frac {9 b e n r x^{3} x^{r} \log {\relax (x )}}{9 r^{2} + 54 r + 81} + \frac {27 b e n x^{3} x^{r} \log {\relax (x )}}{9 r^{2} + 54 r + 81} - \frac {9 b e n x^{3} x^{r}}{9 r^{2} + 54 r + 81} + \frac {9 b e r x^{3} x^{r} \log {\relax (c )}}{9 r^{2} + 54 r + 81} + \frac {27 b e x^{3} x^{r} \log {\relax (c )}}{9 r^{2} + 54 r + 81} & \text {for}\: r \neq -3 \\\frac {a d x^{3}}{3} + a e \log {\relax (x )} + \frac {b d n x^{3} \log {\relax (x )}}{3} - \frac {b d n x^{3}}{9} + \frac {b d x^{3} \log {\relax (c )}}{3} + \frac {b e n \log {\relax (x )}^{2}}{2} + b e \log {\relax (c )} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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