Optimal. Leaf size=122 \[ d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b d^3 n \log ^2(x)-3 b d^2 e n x-\frac {3}{4} b d e^2 n x^2-\frac {1}{9} b e^3 n x^3 \]
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Rubi [A] time = 0.09, antiderivative size = 94, normalized size of antiderivative = 0.77, 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 = {43, 2334, 2301} \[ \frac {1}{6} \left (18 d^2 e x+6 d^3 \log (x)+9 d e^2 x^2+2 e^3 x^3\right ) \left (a+b \log \left (c x^n\right )\right )-3 b d^2 e n x-\frac {1}{2} b d^3 n \log ^2(x)-\frac {3}{4} b d e^2 n x^2-\frac {1}{9} b e^3 n x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2301
Rule 2334
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {1}{6} \left (18 d^2 e x+9 d e^2 x^2+2 e^3 x^3+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {1}{6} e \left (18 d^2+9 d e x+2 e^2 x^2\right )+\frac {d^3 \log (x)}{x}\right ) \, dx\\ &=\frac {1}{6} \left (18 d^2 e x+9 d e^2 x^2+2 e^3 x^3+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 {1}{6} (b e n) \int \left (18 d^2+9 d e x+2 e^2 x^2\right ) \, dx\\ &=-3 b d^2 e n x-\frac {3}{4} b d e^2 n x^2-\frac {1}{9} b e^3 n x^3-\frac {1}{2} b d^3 n \log ^2(x)+\frac {1}{6} \left (18 d^2 e x+9 d e^2 x^2+2 e^3 x^3+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 = 0.06, size = 123, normalized size = 1.01 \[ \frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )+3 a d^2 e x+3 b d^2 e x \log \left (c x^n\right )-3 b d^2 e n x-\frac {3}{4} b d e^2 n x^2-\frac {1}{9} b e^3 n x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 149, normalized size = 1.22 \[ \frac {1}{2} \, b d^{3} n \log \relax (x)^{2} - \frac {1}{9} \, {\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3} - \frac {3}{4} \, {\left (b d e^{2} n - 2 \, a d e^{2}\right )} x^{2} - 3 \, {\left (b d^{2} e n - a d^{2} e\right )} x + \frac {1}{6} \, {\left (2 \, b e^{3} x^{3} + 9 \, b d e^{2} x^{2} + 18 \, b d^{2} e x\right )} \log \relax (c) + \frac {1}{6} \, {\left (2 \, b e^{3} n x^{3} + 9 \, b d e^{2} n x^{2} + 18 \, b d^{2} e n x + 6 \, b d^{3} \log \relax (c) + 6 \, a d^{3}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 150, normalized size = 1.23 \[ \frac {1}{3} \, b n x^{3} e^{3} \log \relax (x) + \frac {3}{2} \, b d n x^{2} e^{2} \log \relax (x) + 3 \, b d^{2} n x e \log \relax (x) + \frac {1}{2} \, b d^{3} n \log \relax (x)^{2} - \frac {1}{9} \, b n x^{3} e^{3} - \frac {3}{4} \, b d n x^{2} e^{2} - 3 \, b d^{2} n x e + \frac {1}{3} \, b x^{3} e^{3} \log \relax (c) + \frac {3}{2} \, b d x^{2} e^{2} \log \relax (c) + 3 \, b d^{2} x e \log \relax (c) + b d^{3} \log \relax (c) \log \relax (x) + \frac {1}{3} \, a x^{3} e^{3} + \frac {3}{2} \, a d x^{2} e^{2} + 3 \, a d^{2} x e + a d^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 579, normalized size = 4.75 \[ \frac {3 b d \,e^{2} x^{2} \ln \relax (c )}{2}+3 b \,d^{2} e x \ln \relax (c )-\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 a d \,e^{2} x^{2}}{2}+3 a \,d^{2} e x +\frac {a \,e^{3} x^{3}}{3}+\frac {b \,e^{3} x^{3} \ln \relax (c )}{3}+\left (\frac {b \,e^{3} x^{3}}{3}+\frac {3 b d \,e^{2} x^{2}}{2}+b \,d^{3} \ln \relax (x )+3 b \,d^{2} e x \right ) \ln \left (x^{n}\right )+b \,d^{3} \ln \relax (c ) \ln \relax (x )+a \,d^{3} \ln \relax (x )-\frac {3 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}-\frac {3 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4}-\frac {i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}+\frac {3 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {3 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {b \,e^{3} n \,x^{3}}{9}+\frac {3 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2}+\frac {i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {3 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4}+\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 {3 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {b \,d^{3} n \ln \relax (x )^{2}}{2}-3 b \,d^{2} e n x -\frac {3 b d \,e^{2} n \,x^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 127, normalized size = 1.04 \[ -\frac {1}{9} \, b e^{3} n x^{3} + \frac {1}{3} \, b e^{3} x^{3} \log \left (c x^{n}\right ) - \frac {3}{4} \, b d e^{2} n x^{2} + \frac {1}{3} \, a e^{3} x^{3} + \frac {3}{2} \, b d e^{2} x^{2} \log \left (c x^{n}\right ) - 3 \, b d^{2} e n x + \frac {3}{2} \, a d e^{2} x^{2} + 3 \, b d^{2} e x \log \left (c x^{n}\right ) + 3 \, a d^{2} e x + \frac {b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.64, size = 106, normalized size = 0.87 \[ \ln \left (c\,x^n\right )\,\left (3\,b\,d^2\,e\,x+\frac {3\,b\,d\,e^2\,x^2}{2}+\frac {b\,e^3\,x^3}{3}\right )+\frac {e^3\,x^3\,\left (3\,a-b\,n\right )}{9}+a\,d^3\,\ln \relax (x)+\frac {b\,d^3\,{\ln \left (c\,x^n\right )}^2}{2\,n}+\frac {3\,d\,e^2\,x^2\,\left (2\,a-b\,n\right )}{4}+3\,d^2\,e\,x\,\left (a-b\,n\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.86, size = 199, normalized size = 1.63 \[ a d^{3} \log {\relax (x )} + 3 a d^{2} e x + \frac {3 a d e^{2} x^{2}}{2} + \frac {a e^{3} x^{3}}{3} + \frac {b d^{3} n \log {\relax (x )}^{2}}{2} + b d^{3} \log {\relax (c )} \log {\relax (x )} + 3 b d^{2} e n x \log {\relax (x )} - 3 b d^{2} e n x + 3 b d^{2} e x \log {\relax (c )} + \frac {3 b d e^{2} n x^{2} \log {\relax (x )}}{2} - \frac {3 b d e^{2} n x^{2}}{4} + \frac {3 b d e^{2} x^{2} \log {\relax (c )}}{2} + \frac {b e^{3} n x^{3} \log {\relax (x )}}{3} - \frac {b e^{3} n x^{3}}{9} + \frac {b e^{3} x^{3} \log {\relax (c )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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