Optimal. Leaf size=152 \[ d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}-\frac {1}{2} b d^3 n \log ^2(x)-\frac {3 b d^2 e n x^r}{r^2}-\frac {3 b d e^2 n x^{2 r}}{4 r^2}-\frac {b e^3 n x^{3 r}}{9 r^2} \]
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Rubi [A] time = 0.17, antiderivative size = 124, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {266, 43, 2334, 12, 14, 2301} \[ \frac {1}{6} \left (\frac {18 d^2 e x^r}{r}+6 d^3 \log (x)+\frac {9 d e^2 x^{2 r}}{r}+\frac {2 e^3 x^{3 r}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n x^r}{r^2}-\frac {1}{2} b d^3 n \log ^2(x)-\frac {3 b d e^2 n x^{2 r}}{4 r^2}-\frac {b e^3 n x^{3 r}}{9 r^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 266
Rule 2301
Rule 2334
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {1}{6} \left (\frac {18 d^2 e x^r}{r}+\frac {9 d e^2 x^{2 r}}{r}+\frac {2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {e x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )+6 d^3 r \log (x)}{6 r x} \, dx\\ &=\frac {1}{6} \left (\frac {18 d^2 e x^r}{r}+\frac {9 d e^2 x^{2 r}}{r}+\frac {2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \frac {e x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )+6 d^3 r \log (x)}{x} \, dx}{6 r}\\ &=\frac {1}{6} \left (\frac {18 d^2 e x^r}{r}+\frac {9 d e^2 x^{2 r}}{r}+\frac {2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \left (18 d^2 e x^{-1+r}+9 d e^2 x^{-1+2 r}+2 e^3 x^{-1+3 r}+\frac {6 d^3 r \log (x)}{x}\right ) \, dx}{6 r}\\ &=-\frac {3 b d^2 e n x^r}{r^2}-\frac {3 b d e^2 n x^{2 r}}{4 r^2}-\frac {b e^3 n x^{3 r}}{9 r^2}+\frac {1}{6} \left (\frac {18 d^2 e x^r}{r}+\frac {9 d e^2 x^{2 r}}{r}+\frac {2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^3 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {3 b d^2 e n x^r}{r^2}-\frac {3 b d e^2 n x^{2 r}}{4 r^2}-\frac {b e^3 n x^{3 r}}{9 r^2}-\frac {1}{2} b d^3 n \log ^2(x)+\frac {1}{6} \left (\frac {18 d^2 e x^r}{r}+\frac {9 d e^2 x^{2 r}}{r}+\frac {2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.37, size = 132, normalized size = 0.87 \[ \frac {1}{36} \left (\frac {e x^r \left (6 a r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )-b n \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )\right )}{r^2}+\frac {18 b d^3 \log ^2\left (c x^n\right )}{n}+\frac {6 b e x^r \log \left (c x^n\right ) \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )}{r}\right )+a d^3 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 169, normalized size = 1.11 \[ \frac {18 \, b d^{3} n r^{2} \log \relax (x)^{2} + 4 \, {\left (3 \, b e^{3} n r \log \relax (x) + 3 \, b e^{3} r \log \relax (c) - b e^{3} n + 3 \, a e^{3} r\right )} x^{3 \, r} + 27 \, {\left (2 \, b d e^{2} n r \log \relax (x) + 2 \, b d e^{2} r \log \relax (c) - b d e^{2} n + 2 \, a d e^{2} r\right )} x^{2 \, r} + 108 \, {\left (b d^{2} e n r \log \relax (x) + b d^{2} e r \log \relax (c) - b d^{2} e n + a d^{2} e r\right )} x^{r} + 36 \, {\left (b d^{3} r^{2} \log \relax (c) + a d^{3} r^{2}\right )} \log \relax (x)}{36 \, r^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 210, normalized size = 1.38 \[ \frac {1}{2} \, b d^{3} n \log \relax (x)^{2} + \frac {3 \, b d^{2} n x^{r} e \log \relax (x)}{r} + b d^{3} \log \relax (c) \log \relax (x) + \frac {3 \, b d^{2} x^{r} e \log \relax (c)}{r} + a d^{3} \log \relax (x) + \frac {3 \, b d n x^{2 \, r} e^{2} \log \relax (x)}{2 \, r} - \frac {3 \, b d^{2} n x^{r} e}{r^{2}} + \frac {3 \, a d^{2} x^{r} e}{r} + \frac {3 \, b d x^{2 \, r} e^{2} \log \relax (c)}{2 \, r} + \frac {b n x^{3 \, r} e^{3} \log \relax (x)}{3 \, r} - \frac {3 \, b d n x^{2 \, r} e^{2}}{4 \, r^{2}} + \frac {3 \, a d x^{2 \, r} e^{2}}{2 \, r} + \frac {b x^{3 \, r} e^{3} \log \relax (c)}{3 \, r} - \frac {b n x^{3 \, r} e^{3}}{9 \, r^{2}} + \frac {a x^{3 \, r} e^{3}}{3 \, r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 693, normalized size = 4.56 \[ \frac {a \,e^{3} x^{3 r}}{3 r}+\frac {\left (6 d^{3} r \ln \relax (x )+18 d^{2} e \,x^{r}+9 d \,e^{2} x^{2 r}+2 e^{3} x^{3 r}\right ) b \ln \left (x^{n}\right )}{6 r}+b \,d^{3} \ln \relax (c ) \ln \relax (x )+a \,d^{3} \ln \relax (x )-\frac {i \pi b \,d^{3} \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 {3 i \pi b \,d^{2} e \,x^{r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 r}-\frac {3 i \pi b d \,e^{2} x^{2 r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 r}+\frac {i \pi b \,e^{3} x^{3 r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6 r}+\frac {i \pi b \,e^{3} x^{3 r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6 r}+\frac {3 b d \,e^{2} x^{2 r} \ln \relax (c )}{2 r}+\frac {3 b \,d^{2} e \,x^{r} \ln \relax (c )}{r}-\frac {3 i \pi b \,d^{2} 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 {3 i \pi b d \,e^{2} x^{2 r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 r}+\frac {3 a \,d^{2} e \,x^{r}}{r}+\frac {b \,e^{3} x^{3 r} \ln \relax (c )}{3 r}+\frac {3 a d \,e^{2} x^{2 r}}{2 r}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2}-\frac {3 b \,d^{2} e n \,x^{r}}{r^{2}}-\frac {3 b d \,e^{2} n \,x^{2 r}}{4 r^{2}}-\frac {b \,d^{3} n \ln \relax (x )^{2}}{2}-\frac {b \,e^{3} n \,x^{3 r}}{9 r^{2}}+\frac {3 i \pi b \,d^{2} e \,x^{r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 r}+\frac {3 i \pi b \,d^{2} e \,x^{r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 r}+\frac {3 i \pi b d \,e^{2} x^{2 r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 r}+\frac {3 i \pi b d \,e^{2} x^{2 r} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 r}-\frac {i \pi b \,e^{3} x^{3 r} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6 r}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi b \,e^{3} x^{3 r} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6 r} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 172, normalized size = 1.13 \[ \frac {b e^{3} x^{3 \, r} \log \left (c x^{n}\right )}{3 \, r} + \frac {3 \, b d e^{2} x^{2 \, r} \log \left (c x^{n}\right )}{2 \, r} + \frac {3 \, b d^{2} e x^{r} \log \left (c x^{n}\right )}{r} + \frac {b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \relax (x) - \frac {b e^{3} n x^{3 \, r}}{9 \, r^{2}} + \frac {a e^{3} x^{3 \, r}}{3 \, r} - \frac {3 \, b d e^{2} n x^{2 \, r}}{4 \, r^{2}} + \frac {3 \, a d e^{2} x^{2 \, r}}{2 \, r} - \frac {3 \, b d^{2} e n x^{r}}{r^{2}} + \frac {3 \, a d^{2} e x^{r}}{r} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.73, size = 286, normalized size = 1.88 \[ \begin {cases} a d^{3} \log {\relax (x )} + \frac {3 a d^{2} e x^{r}}{r} + \frac {3 a d e^{2} x^{2 r}}{2 r} + \frac {a e^{3} x^{3 r}}{3 r} + \frac {b d^{3} n \log {\relax (x )}^{2}}{2} + b d^{3} \log {\relax (c )} \log {\relax (x )} + \frac {3 b d^{2} e n x^{r} \log {\relax (x )}}{r} - \frac {3 b d^{2} e n x^{r}}{r^{2}} + \frac {3 b d^{2} e x^{r} \log {\relax (c )}}{r} + \frac {3 b d e^{2} n x^{2 r} \log {\relax (x )}}{2 r} - \frac {3 b d e^{2} n x^{2 r}}{4 r^{2}} + \frac {3 b d e^{2} x^{2 r} \log {\relax (c )}}{2 r} + \frac {b e^{3} n x^{3 r} \log {\relax (x )}}{3 r} - \frac {b e^{3} n x^{3 r}}{9 r^{2}} + \frac {b e^{3} x^{3 r} \log {\relax (c )}}{3 r} & \text {for}\: r \neq 0 \\\left (d + e\right )^{3} \left (\begin {cases} a \log {\relax (x )} & \text {for}\: b = 0 \\- \left (- a - b \log {\relax (c )}\right ) \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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