Optimal. Leaf size=57 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b d n}{9 x^3}-\frac {b e n}{4 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 48, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {43, 2334, 12} \[ -\frac {1}{6} \left (\frac {2 d}{x^3}+\frac {3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{9 x^3}-\frac {b e n}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2334
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{6} \left (\frac {2 d}{x^3}+\frac {3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-2 d-3 e x}{6 x^4} \, dx\\ &=-\frac {1}{6} \left (\frac {2 d}{x^3}+\frac {3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (b n) \int \frac {-2 d-3 e x}{x^4} \, dx\\ &=-\frac {1}{6} \left (\frac {2 d}{x^3}+\frac {3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (b n) \int \left (-\frac {2 d}{x^4}-\frac {3 e}{x^3}\right ) \, dx\\ &=-\frac {b d n}{9 x^3}-\frac {b e n}{4 x^2}-\frac {1}{6} \left (\frac {2 d}{x^3}+\frac {3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.82 \[ -\frac {6 a (2 d+3 e x)+6 b (2 d+3 e x) \log \left (c x^n\right )+b n (4 d+9 e x)}{36 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 57, normalized size = 1.00 \[ -\frac {4 \, b d n + 12 \, a d + 9 \, {\left (b e n + 2 \, a e\right )} x + 6 \, {\left (3 \, b e x + 2 \, b d\right )} \log \relax (c) + 6 \, {\left (3 \, b e n x + 2 \, b d n\right )} \log \relax (x)}{36 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 58, normalized size = 1.02 \[ -\frac {18 \, b n x e \log \relax (x) + 9 \, b n x e + 18 \, b x e \log \relax (c) + 12 \, b d n \log \relax (x) + 4 \, b d n + 18 \, a x e + 12 \, b d \log \relax (c) + 12 \, a d}{36 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 235, normalized size = 4.12 \[ -\frac {\left (3 e x +2 d \right ) b \ln \left (x^{n}\right )}{6 x^{3}}-\frac {-9 i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+9 i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-6 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+6 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+9 b e n x +18 b e x \ln \relax (c )+18 a e x +4 b d n +12 b d \ln \relax (c )+12 a d}{36 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 57, normalized size = 1.00 \[ -\frac {b e n}{4 \, x^{2}} - \frac {b e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {b d n}{9 \, x^{3}} - \frac {a e}{2 \, x^{2}} - \frac {b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 49, normalized size = 0.86 \[ -\frac {2\,a\,d+x\,\left (3\,a\,e+\frac {3\,b\,e\,n}{2}\right )+\frac {2\,b\,d\,n}{3}}{6\,x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d}{3}+\frac {b\,e\,x}{2}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.64, size = 88, normalized size = 1.54 \[ - \frac {a d}{3 x^{3}} - \frac {a e}{2 x^{2}} - \frac {b d n \log {\relax (x )}}{3 x^{3}} - \frac {b d n}{9 x^{3}} - \frac {b d \log {\relax (c )}}{3 x^{3}} - \frac {b e n \log {\relax (x )}}{2 x^{2}} - \frac {b e n}{4 x^{2}} - \frac {b e \log {\relax (c )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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