Optimal. Leaf size=57 \[ -\frac {b d n}{9 x^3}-\frac {b e n}{4 x^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {45, 2372, 12}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2372
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 = 0.02, size = 47, normalized size = 0.82 \begin {gather*} -\frac {6 a (2 d+3 e x)+b n (4 d+9 e x)+6 b (2 d+3 e x) \log \left (c x^n\right )}{36 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.06, size = 235, normalized size = 4.12
method | result | size |
risch | \(-\frac {b \left (3 e x +2 d \right ) \ln \left (x^{n}\right )}{6 x^{3}}-\frac {-9 i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+9 i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 \ln \left (c \right ) b e x +9 b e n x +18 a e x -6 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+6 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+12 d b \ln \left (c \right )+4 b d n +12 a d}{36 x^{3}}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 60, normalized size = 1.05 \begin {gather*} -\frac {b n e}{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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 59, normalized size = 1.04 \begin {gather*} -\frac {4 \, b d n + 9 \, {\left (b n + 2 \, a\right )} x e + 12 \, a d + 6 \, {\left (3 \, b x e + 2 \, b d\right )} \log \left (c\right ) + 6 \, {\left (3 \, b n x e + 2 \, b d n\right )} \log \left (x\right )}{36 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 68, normalized size = 1.19 \begin {gather*} - \frac {a d}{3 x^{3}} - \frac {a e}{2 x^{2}} - \frac {b d n}{9 x^{3}} - \frac {b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {b e n}{4 x^{2}} - \frac {b e \log {\left (c x^{n} \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.85, size = 58, normalized size = 1.02 \begin {gather*} -\frac {18 \, b n x e \log \left (x\right ) + 9 \, b n x e + 18 \, b x e \log \left (c\right ) + 12 \, b d n \log \left (x\right ) + 4 \, b d n + 18 \, a x e + 12 \, b d \log \left (c\right ) + 12 \, a d}{36 \, x^{3}} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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