Optimal. Leaf size=142 \[ \frac {e (d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{20 d^2 x^4}-\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}-\frac {b e^5 n \log (x)}{20 d^2}-\frac {b n (d+e x)^5}{25 d^2 x^5}+\frac {b d^2 e n}{80 x^4}+\frac {b e^4 n}{5 d x}+\frac {b d e^2 n}{15 x^3}+\frac {3 b e^3 n}{20 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 133, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {45, 37, 2334, 12, 78, 43} \[ -\frac {1}{20} \left (\frac {4 (d+e x)^4}{d x^5}-\frac {e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b e^5 n \log (x)}{20 d^2}+\frac {b d^2 e n}{80 x^4}-\frac {b n (d+e x)^5}{25 d^2 x^5}+\frac {b d e^2 n}{15 x^3}+\frac {b e^4 n}{5 d x}+\frac {3 b e^3 n}{20 x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 43
Rule 45
Rule 78
Rule 2334
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac {1}{20} \left (\frac {4 (d+e x)^4}{d x^5}-\frac {e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {(-4 d+e x) (d+e x)^4}{20 d^2 x^6} \, dx\\ &=-\frac {1}{20} \left (\frac {4 (d+e x)^4}{d x^5}-\frac {e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \frac {(-4 d+e x) (d+e x)^4}{x^6} \, dx}{20 d^2}\\ &=-\frac {b n (d+e x)^5}{25 d^2 x^5}-\frac {1}{20} \left (\frac {4 (d+e x)^4}{d x^5}-\frac {e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b e n) \int \frac {(d+e x)^4}{x^5} \, dx}{20 d^2}\\ &=-\frac {b n (d+e x)^5}{25 d^2 x^5}-\frac {1}{20} \left (\frac {4 (d+e x)^4}{d x^5}-\frac {e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b e n) \int \left (\frac {d^4}{x^5}+\frac {4 d^3 e}{x^4}+\frac {6 d^2 e^2}{x^3}+\frac {4 d e^3}{x^2}+\frac {e^4}{x}\right ) \, dx}{20 d^2}\\ &=\frac {b d^2 e n}{80 x^4}+\frac {b d e^2 n}{15 x^3}+\frac {3 b e^3 n}{20 x^2}+\frac {b e^4 n}{5 d x}-\frac {b n (d+e x)^5}{25 d^2 x^5}-\frac {b e^5 n \log (x)}{20 d^2}-\frac {1}{20} \left (\frac {4 (d+e x)^4}{d x^5}-\frac {e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 113, normalized size = 0.80 \[ -\frac {60 a \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right )+60 b \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (48 d^3+225 d^2 e x+400 d e^2 x^2+300 e^3 x^3\right )}{1200 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 155, normalized size = 1.09 \[ -\frac {48 \, b d^{3} n + 240 \, a d^{3} + 300 \, {\left (b e^{3} n + 2 \, a e^{3}\right )} x^{3} + 400 \, {\left (b d e^{2} n + 3 \, a d e^{2}\right )} x^{2} + 225 \, {\left (b d^{2} e n + 4 \, a d^{2} e\right )} x + 60 \, {\left (10 \, b e^{3} x^{3} + 20 \, b d e^{2} x^{2} + 15 \, b d^{2} e x + 4 \, b d^{3}\right )} \log \relax (c) + 60 \, {\left (10 \, b e^{3} n x^{3} + 20 \, b d e^{2} n x^{2} + 15 \, b d^{2} e n x + 4 \, b d^{3} n\right )} \log \relax (x)}{1200 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 158, normalized size = 1.11 \[ -\frac {600 \, b n x^{3} e^{3} \log \relax (x) + 1200 \, b d n x^{2} e^{2} \log \relax (x) + 900 \, b d^{2} n x e \log \relax (x) + 300 \, b n x^{3} e^{3} + 400 \, b d n x^{2} e^{2} + 225 \, b d^{2} n x e + 600 \, b x^{3} e^{3} \log \relax (c) + 1200 \, b d x^{2} e^{2} \log \relax (c) + 900 \, b d^{2} x e \log \relax (c) + 240 \, b d^{3} n \log \relax (x) + 48 \, b d^{3} n + 600 \, a x^{3} e^{3} + 1200 \, a d x^{2} e^{2} + 900 \, a d^{2} x e + 240 \, b d^{3} \log \relax (c) + 240 \, a d^{3}}{1200 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 571, normalized size = 4.02 \[ -\frac {\left (10 e^{3} x^{3}+20 d \,e^{2} x^{2}+15 d^{2} e x +4 d^{3}\right ) b \ln \left (x^{n}\right )}{20 x^{5}}-\frac {1200 b d \,e^{2} x^{2} \ln \relax (c )+900 b \,d^{2} e x \ln \relax (c )+1200 a d \,e^{2} x^{2}+900 a \,d^{2} e x +240 a \,d^{3}+600 a \,e^{3} x^{3}+48 b \,d^{3} n +240 b \,d^{3} \ln \relax (c )+600 b \,e^{3} x^{3} \ln \relax (c )-450 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 )-600 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 )-120 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 )-300 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 )+600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+300 b \,e^{3} n \,x^{3}-120 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+450 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+450 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-450 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+120 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+120 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+225 b \,d^{2} e n x +400 b d \,e^{2} n \,x^{2}}{1200 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 143, normalized size = 1.01 \[ -\frac {b e^{3} n}{4 \, x^{2}} - \frac {b e^{3} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {b d e^{2} n}{3 \, x^{3}} - \frac {a e^{3}}{2 \, x^{2}} - \frac {b d e^{2} \log \left (c x^{n}\right )}{x^{3}} - \frac {3 \, b d^{2} e n}{16 \, x^{4}} - \frac {a d e^{2}}{x^{3}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {b d^{3} n}{25 \, x^{5}} - \frac {3 \, a d^{2} e}{4 \, x^{4}} - \frac {b d^{3} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d^{3}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 120, normalized size = 0.85 \[ -\frac {x^3\,\left (10\,a\,e^3+5\,b\,e^3\,n\right )+x\,\left (15\,a\,d^2\,e+\frac {15\,b\,d^2\,e\,n}{4}\right )+4\,a\,d^3+x^2\,\left (20\,a\,d\,e^2+\frac {20\,b\,d\,e^2\,n}{3}\right )+\frac {4\,b\,d^3\,n}{5}}{20\,x^5}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{5}+\frac {3\,b\,d^2\,e\,x}{4}+b\,d\,e^2\,x^2+\frac {b\,e^3\,x^3}{2}\right )}{x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.65, size = 219, normalized size = 1.54 \[ - \frac {a d^{3}}{5 x^{5}} - \frac {3 a d^{2} e}{4 x^{4}} - \frac {a d e^{2}}{x^{3}} - \frac {a e^{3}}{2 x^{2}} - \frac {b d^{3} n \log {\relax (x )}}{5 x^{5}} - \frac {b d^{3} n}{25 x^{5}} - \frac {b d^{3} \log {\relax (c )}}{5 x^{5}} - \frac {3 b d^{2} e n \log {\relax (x )}}{4 x^{4}} - \frac {3 b d^{2} e n}{16 x^{4}} - \frac {3 b d^{2} e \log {\relax (c )}}{4 x^{4}} - \frac {b d e^{2} n \log {\relax (x )}}{x^{3}} - \frac {b d e^{2} n}{3 x^{3}} - \frac {b d e^{2} \log {\relax (c )}}{x^{3}} - \frac {b e^{3} n \log {\relax (x )}}{2 x^{2}} - \frac {b e^{3} n}{4 x^{2}} - \frac {b e^{3} \log {\relax (c )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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