Optimal. Leaf size=59 \[ -\frac {1}{16} b d n x^4-\frac {b e n x^{4+r}}{(4+r)^2}+\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A]
time = 0.06, 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, 2371, 12}
\begin {gather*} \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d n x^4-\frac {b e n x^{r+4}}{(r+4)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2371
Rubi steps
\begin {align*} \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{4} x^3 \left (d+\frac {4 e x^r}{4+r}\right ) \, dx\\ &=\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int x^3 \left (d+\frac {4 e x^r}{4+r}\right ) \, dx\\ &=\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (d x^3+\frac {4 e x^{3+r}}{4+r}\right ) \, dx\\ &=-\frac {1}{16} b d n x^4-\frac {b e n x^{4+r}}{(4+r)^2}+\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 73, normalized size = 1.24 \begin {gather*} \frac {x^4 \left (4 a (4+r) \left (d (4+r)+4 e x^r\right )-b n \left (d (4+r)^2+16 e x^r\right )+4 b (4+r) \left (d (4+r)+4 e x^r\right ) \log \left (c x^n\right )\right )}{16 (4+r)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.08, size = 613, normalized size = 10.39
method | result | size |
risch | \(\frac {b \,x^{4} \left (d r +4 e \,x^{r}+4 d \right ) \ln \left (x^{n}\right )}{16+4 r}-\frac {x^{4} \left (-64 x^{r} a e +16 b d n +16 x^{r} b e n -16 x^{r} a e r -64 a d +8 b d n r -32 \ln \left (c \right ) b d r -4 \ln \left (c \right ) b d \,r^{2}-4 a d \,r^{2}-64 d b \ln \left (c \right )-16 \ln \left (c \right ) b e \,x^{r} r -32 a d r -2 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-32 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-32 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-2 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-16 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -16 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r +32 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b d n \,r^{2}+32 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r}-8 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -8 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +2 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 )+16 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 ) r -64 \ln \left (c \right ) b e \,x^{r}+8 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r} r +16 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} r +2 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+32 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+32 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 )-32 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-32 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r \right )}{16 \left (4+r \right )^{2}}\) | \(613\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 1.29 \begin {gather*} -\frac {1}{16} \, b d n x^{4} + \frac {1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d x^{4} + \frac {b e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {a e x^{r + 4}}{r + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs.
\(2 (57) = 114\).
time = 0.35, size = 158, normalized size = 2.68 \begin {gather*} \frac {4 \, {\left (b d r^{2} + 8 \, b d r + 16 \, b d\right )} x^{4} \log \left (c\right ) + 4 \, {\left (b d n r^{2} + 8 \, b d n r + 16 \, b d n\right )} x^{4} \log \left (x\right ) - {\left (16 \, b d n + {\left (b d n - 4 \, a d\right )} r^{2} - 64 \, a d + 8 \, {\left (b d n - 4 \, a d\right )} r\right )} x^{4} + 16 \, {\left ({\left (b r + 4 \, b\right )} x^{4} e \log \left (c\right ) + {\left (b n r + 4 \, b n\right )} x^{4} e \log \left (x\right ) - {\left (b n - a r - 4 \, a\right )} x^{4} e\right )} x^{r}}{16 \, {\left (r^{2} + 8 \, r + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs.
\(2 (51) = 102\).
time = 3.84, size = 398, normalized size = 6.75 \begin {gather*} \begin {cases} \frac {4 a d r^{2} x^{4}}{16 r^{2} + 128 r + 256} + \frac {32 a d r x^{4}}{16 r^{2} + 128 r + 256} + \frac {64 a d x^{4}}{16 r^{2} + 128 r + 256} + \frac {16 a e r x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {64 a e x^{4} x^{r}}{16 r^{2} + 128 r + 256} - \frac {b d n r^{2} x^{4}}{16 r^{2} + 128 r + 256} - \frac {8 b d n r x^{4}}{16 r^{2} + 128 r + 256} - \frac {16 b d n x^{4}}{16 r^{2} + 128 r + 256} + \frac {4 b d r^{2} x^{4} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} + \frac {32 b d r x^{4} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} + \frac {64 b d x^{4} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} - \frac {16 b e n x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {16 b e r x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} + \frac {64 b e x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} & \text {for}\: r \neq -4 \\\frac {a d x^{4}}{4} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{4}}{16} + \frac {b d x^{4} \log {\left (c x^{n} \right )}}{4} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 137 vs.
\(2 (57) = 114\).
time = 3.32, size = 137, normalized size = 2.32 \begin {gather*} \frac {b n r x^{4} x^{r} e \log \left (x\right )}{r^{2} + 8 \, r + 16} + \frac {1}{4} \, b d n x^{4} \log \left (x\right ) + \frac {4 \, b n x^{4} x^{r} e \log \left (x\right )}{r^{2} + 8 \, r + 16} - \frac {1}{16} \, b d n x^{4} - \frac {b n x^{4} x^{r} e}{r^{2} + 8 \, r + 16} + \frac {1}{4} \, b d x^{4} \log \left (c\right ) + \frac {b x^{4} x^{r} e \log \left (c\right )}{r + 4} + \frac {1}{4} \, a d x^{4} + \frac {a x^{4} x^{r} e}{r + 4} \end {gather*}
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 \begin {gather*} \int x^3\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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