Optimal. Leaf size=118 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+e^3 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{4 x^2}-\frac {3 b d^2 e n}{x}-\frac {3}{2} b d e^2 n \log ^2(x)-b e^3 n x \]
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Rubi [A] time = 0.09, antiderivative size = 91, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ -\frac {1}{2} \left (\frac {6 d^2 e}{x}+\frac {d^3}{x^2}-6 d e^2 \log (x)-2 e^3 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n}{x}-\frac {b d^3 n}{4 x^2}-\frac {3}{2} b d e^2 n \log ^2(x)-b e^3 n x \]
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^3} \, dx &=-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {6 d^2 e}{x}-2 e^3 x-6 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^3-\frac {d^3}{2 x^3}-\frac {3 d^2 e}{x^2}+\frac {3 d e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac {b d^3 n}{4 x^2}-\frac {3 b d^2 e n}{x}-b e^3 n x-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {6 d^2 e}{x}-2 e^3 x-6 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (3 b d e^2 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {b d^3 n}{4 x^2}-\frac {3 b d^2 e n}{x}-b e^3 n x-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{2} \left (\frac {d^3}{x^2}+\frac {6 d^2 e}{x}-2 e^3 x-6 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 115, normalized size = 0.97 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+a e^3 x+b e^3 x \log \left (c x^n\right )-\frac {b d^3 n}{4 x^2}-\frac {3 b d^2 e n}{x}-b e^3 n x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 150, normalized size = 1.27 \[ \frac {6 \, b d e^{2} n x^{2} \log \relax (x)^{2} - b d^{3} n - 2 \, a d^{3} - 4 \, {\left (b e^{3} n - a e^{3}\right )} x^{3} - 12 \, {\left (b d^{2} e n + a d^{2} e\right )} x + 2 \, {\left (2 \, b e^{3} x^{3} - 6 \, b d^{2} e x - b d^{3}\right )} \log \relax (c) + 2 \, {\left (2 \, b e^{3} n x^{3} + 6 \, b d e^{2} x^{2} \log \relax (c) - 6 \, b d^{2} e n x + 6 \, a d e^{2} x^{2} - b d^{3} n\right )} \log \relax (x)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 154, normalized size = 1.31 \[ \frac {6 \, b d n x^{2} e^{2} \log \relax (x)^{2} + 4 \, b n x^{3} e^{3} \log \relax (x) - 12 \, b d^{2} n x e \log \relax (x) + 12 \, b d x^{2} e^{2} \log \relax (c) \log \relax (x) - 4 \, b n x^{3} e^{3} - 12 \, b d^{2} n x e + 4 \, b x^{3} e^{3} \log \relax (c) - 12 \, b d^{2} x e \log \relax (c) - 2 \, b d^{3} n \log \relax (x) + 12 \, a d x^{2} e^{2} \log \relax (x) - b d^{3} n + 4 \, a x^{3} e^{3} - 12 \, a d^{2} x e - 2 \, b d^{3} \log \relax (c) - 2 \, a d^{3}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 586, normalized size = 4.97 \[ -\frac {\left (-6 d \,e^{2} x^{2} \ln \relax (x )-2 e^{3} x^{3}+6 d^{2} e x +d^{3}\right ) b \ln \left (x^{n}\right )}{2 x^{2}}-\frac {12 b \,d^{2} e x \ln \relax (c )+12 a \,d^{2} e x +2 a \,d^{3}-4 a \,e^{3} x^{3}+b \,d^{3} n +2 b \,d^{3} \ln \relax (c )-4 b \,e^{3} x^{3} \ln \relax (c )-6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )-6 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 )-12 a d \,e^{2} x^{2} \ln \relax (x )+6 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 ) \ln \relax (x )-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 )+2 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 )+4 b \,e^{3} n \,x^{3}-i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )+6 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-12 b d \,e^{2} x^{2} \ln \relax (c ) \ln \relax (x )+6 b d \,e^{2} n \,x^{2} \ln \relax (x )^{2}+2 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )+12 b \,d^{2} e n x}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 125, normalized size = 1.06 \[ -b e^{3} n x + b e^{3} x \log \left (c x^{n}\right ) + a e^{3} x + \frac {3 \, b d e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d e^{2} \log \relax (x) - \frac {3 \, b d^{2} e n}{x} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{x} - \frac {b d^{3} n}{4 \, x^{2}} - \frac {3 \, a d^{2} e}{x} - \frac {b d^{3} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a d^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 139, normalized size = 1.18 \[ \ln \relax (x)\,\left (3\,a\,d\,e^2+\frac {9\,b\,d\,e^2\,n}{2}\right )-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^3}{2}+3\,b\,d^2\,e\,x+\frac {9\,b\,d\,e^2\,x^2}{2}+2\,b\,e^3\,x^3}{x^2}-3\,b\,e^3\,x\right )-\frac {x\,\left (6\,a\,d^2\,e+6\,b\,d^2\,e\,n\right )+a\,d^3+\frac {b\,d^3\,n}{2}}{2\,x^2}+e^3\,x\,\left (a-b\,n\right )+\frac {3\,b\,d\,e^2\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.97, size = 182, normalized size = 1.54 \[ - \frac {a d^{3}}{2 x^{2}} - \frac {3 a d^{2} e}{x} + 3 a d e^{2} \log {\relax (x )} + a e^{3} x - \frac {b d^{3} n \log {\relax (x )}}{2 x^{2}} - \frac {b d^{3} n}{4 x^{2}} - \frac {b d^{3} \log {\relax (c )}}{2 x^{2}} - \frac {3 b d^{2} e n \log {\relax (x )}}{x} - \frac {3 b d^{2} e n}{x} - \frac {3 b d^{2} e \log {\relax (c )}}{x} + \frac {3 b d e^{2} n \log {\relax (x )}^{2}}{2} + 3 b d e^{2} \log {\relax (c )} \log {\relax (x )} + b e^{3} n x \log {\relax (x )} - b e^{3} n x + b e^{3} x \log {\relax (c )} \]
Verification of antiderivative is not currently implemented for this CAS.
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