Optimal. Leaf size=133 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {b d^3 n}{36 x^6}-\frac {3 b d^2 e n}{25 x^5}-\frac {3 b d e^2 n}{16 x^4}-\frac {b e^3 n}{9 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 100, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {43, 2334, 12, 14} \[ -\frac {1}{60} \left (\frac {36 d^2 e}{x^5}+\frac {10 d^3}{x^6}+\frac {45 d e^2}{x^4}+\frac {20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n}{25 x^5}-\frac {b d^3 n}{36 x^6}-\frac {3 b d e^2 n}{16 x^4}-\frac {b e^3 n}{9 x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 2334
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx &=-\frac {1}{60} \left (\frac {10 d^3}{x^6}+\frac {36 d^2 e}{x^5}+\frac {45 d e^2}{x^4}+\frac {20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-10 d^3-36 d^2 e x-45 d e^2 x^2-20 e^3 x^3}{60 x^7} \, dx\\ &=-\frac {1}{60} \left (\frac {10 d^3}{x^6}+\frac {36 d^2 e}{x^5}+\frac {45 d e^2}{x^4}+\frac {20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int \frac {-10 d^3-36 d^2 e x-45 d e^2 x^2-20 e^3 x^3}{x^7} \, dx\\ &=-\frac {1}{60} \left (\frac {10 d^3}{x^6}+\frac {36 d^2 e}{x^5}+\frac {45 d e^2}{x^4}+\frac {20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} (b n) \int \left (-\frac {10 d^3}{x^7}-\frac {36 d^2 e}{x^6}-\frac {45 d e^2}{x^5}-\frac {20 e^3}{x^4}\right ) \, dx\\ &=-\frac {b d^3 n}{36 x^6}-\frac {3 b d^2 e n}{25 x^5}-\frac {3 b d e^2 n}{16 x^4}-\frac {b e^3 n}{9 x^3}-\frac {1}{60} \left (\frac {10 d^3}{x^6}+\frac {36 d^2 e}{x^5}+\frac {45 d e^2}{x^4}+\frac {20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 113, normalized size = 0.85 \[ -\frac {60 a \left (10 d^3+36 d^2 e x+45 d e^2 x^2+20 e^3 x^3\right )+60 b \left (10 d^3+36 d^2 e x+45 d e^2 x^2+20 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (100 d^3+432 d^2 e x+675 d e^2 x^2+400 e^3 x^3\right )}{3600 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 155, normalized size = 1.17 \[ -\frac {100 \, b d^{3} n + 600 \, a d^{3} + 400 \, {\left (b e^{3} n + 3 \, a e^{3}\right )} x^{3} + 675 \, {\left (b d e^{2} n + 4 \, a d e^{2}\right )} x^{2} + 432 \, {\left (b d^{2} e n + 5 \, a d^{2} e\right )} x + 60 \, {\left (20 \, b e^{3} x^{3} + 45 \, b d e^{2} x^{2} + 36 \, b d^{2} e x + 10 \, b d^{3}\right )} \log \relax (c) + 60 \, {\left (20 \, b e^{3} n x^{3} + 45 \, b d e^{2} n x^{2} + 36 \, b d^{2} e n x + 10 \, b d^{3} n\right )} \log \relax (x)}{3600 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 158, normalized size = 1.19 \[ -\frac {1200 \, b n x^{3} e^{3} \log \relax (x) + 2700 \, b d n x^{2} e^{2} \log \relax (x) + 2160 \, b d^{2} n x e \log \relax (x) + 400 \, b n x^{3} e^{3} + 675 \, b d n x^{2} e^{2} + 432 \, b d^{2} n x e + 1200 \, b x^{3} e^{3} \log \relax (c) + 2700 \, b d x^{2} e^{2} \log \relax (c) + 2160 \, b d^{2} x e \log \relax (c) + 600 \, b d^{3} n \log \relax (x) + 100 \, b d^{3} n + 1200 \, a x^{3} e^{3} + 2700 \, a d x^{2} e^{2} + 2160 \, a d^{2} x e + 600 \, b d^{3} \log \relax (c) + 600 \, a d^{3}}{3600 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 571, normalized size = 4.29 \[ -\frac {\left (20 e^{3} x^{3}+45 d \,e^{2} x^{2}+36 d^{2} e x +10 d^{3}\right ) b \ln \left (x^{n}\right )}{60 x^{6}}-\frac {2700 b d \,e^{2} x^{2} \ln \relax (c )+2160 b \,d^{2} e x \ln \relax (c )+2700 a d \,e^{2} x^{2}+2160 a \,d^{2} e x +600 a \,d^{3}+1200 a \,e^{3} x^{3}+100 b \,d^{3} n +600 b \,d^{3} \ln \relax (c )+1200 b \,e^{3} x^{3} \ln \relax (c )-1080 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 )-1350 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 )-300 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 )-600 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 )+1350 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+1350 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+400 b \,e^{3} n \,x^{3}-300 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+1080 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+1080 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-600 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+600 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+600 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-1350 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-1080 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+300 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+432 b \,d^{2} e n x +675 b d \,e^{2} n \,x^{2}}{3600 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 143, normalized size = 1.08 \[ -\frac {b e^{3} n}{9 \, x^{3}} - \frac {b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {3 \, b d e^{2} n}{16 \, x^{4}} - \frac {a e^{3}}{3 \, x^{3}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {3 \, b d^{2} e n}{25 \, x^{5}} - \frac {3 \, a d e^{2}}{4 \, x^{4}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {b d^{3} n}{36 \, x^{6}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {b d^{3} \log \left (c x^{n}\right )}{6 \, x^{6}} - \frac {a d^{3}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 121, normalized size = 0.91 \[ -\frac {x^3\,\left (20\,a\,e^3+\frac {20\,b\,e^3\,n}{3}\right )+x\,\left (36\,a\,d^2\,e+\frac {36\,b\,d^2\,e\,n}{5}\right )+10\,a\,d^3+x^2\,\left (45\,a\,d\,e^2+\frac {45\,b\,d\,e^2\,n}{4}\right )+\frac {5\,b\,d^3\,n}{3}}{60\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{6}+\frac {3\,b\,d^2\,e\,x}{5}+\frac {3\,b\,d\,e^2\,x^2}{4}+\frac {b\,e^3\,x^3}{3}\right )}{x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.12, size = 231, normalized size = 1.74 \[ - \frac {a d^{3}}{6 x^{6}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {3 a d e^{2}}{4 x^{4}} - \frac {a e^{3}}{3 x^{3}} - \frac {b d^{3} n \log {\relax (x )}}{6 x^{6}} - \frac {b d^{3} n}{36 x^{6}} - \frac {b d^{3} \log {\relax (c )}}{6 x^{6}} - \frac {3 b d^{2} e n \log {\relax (x )}}{5 x^{5}} - \frac {3 b d^{2} e n}{25 x^{5}} - \frac {3 b d^{2} e \log {\relax (c )}}{5 x^{5}} - \frac {3 b d e^{2} n \log {\relax (x )}}{4 x^{4}} - \frac {3 b d e^{2} n}{16 x^{4}} - \frac {3 b d e^{2} \log {\relax (c )}}{4 x^{4}} - \frac {b e^{3} n \log {\relax (x )}}{3 x^{3}} - \frac {b e^{3} n}{9 x^{3}} - \frac {b e^{3} \log {\relax (c )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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