Optimal. Leaf size=53 \[ -\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b d n}{9 x^3}-\frac {b e n}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {14, 2334, 12} \[ -\frac {1}{3} \left (\frac {d}{x^3}+\frac {3 e}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{9 x^3}-\frac {b e n}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2334
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d}{x^3}+\frac {3 e}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d-3 e x^2}{3 x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d}{x^3}+\frac {3 e}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \frac {-d-3 e x^2}{x^4} \, dx\\ &=-\frac {1}{3} \left (\frac {d}{x^3}+\frac {3 e}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (-\frac {d}{x^4}-\frac {3 e}{x^2}\right ) \, dx\\ &=-\frac {b d n}{9 x^3}-\frac {b e n}{x}-\frac {1}{3} \left (\frac {d}{x^3}+\frac {3 e}{x}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 63, normalized size = 1.19 \[ -\frac {a d}{3 x^3}-\frac {a e}{x}-\frac {b d \log \left (c x^n\right )}{3 x^3}-\frac {b e \log \left (c x^n\right )}{x}-\frac {b d n}{9 x^3}-\frac {b e n}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 59, normalized size = 1.11 \[ -\frac {b d n + 9 \, {\left (b e n + a e\right )} x^{2} + 3 \, a d + 3 \, {\left (3 \, b e x^{2} + b d\right )} \log \relax (c) + 3 \, {\left (3 \, b e n x^{2} + b d n\right )} \log \relax (x)}{9 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 65, normalized size = 1.23 \[ -\frac {9 \, b n x^{2} e \log \relax (x) + 9 \, b n x^{2} e + 9 \, b x^{2} e \log \relax (c) + 9 \, a x^{2} e + 3 \, b d n \log \relax (x) + b d n + 3 \, b d \log \relax (c) + 3 \, a d}{9 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 249, normalized size = 4.70 \[ -\frac {\left (3 e \,x^{2}+d \right ) b \ln \left (x^{n}\right )}{3 x^{3}}-\frac {-9 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+9 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-3 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 b e n \,x^{2}+18 b e \,x^{2} \ln \relax (c )+18 a e \,x^{2}+2 b d n +6 b d \ln \relax (c )+6 a d}{18 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 57, normalized size = 1.08 \[ -\frac {b e n}{x} - \frac {b e \log \left (c x^{n}\right )}{x} - \frac {a e}{x} - \frac {b d n}{9 \, x^{3}} - \frac {b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 51, normalized size = 0.96 \[ -\frac {\left (3\,a\,e+3\,b\,e\,n\right )\,x^2+a\,d+\frac {b\,d\,n}{3}}{3\,x^3}-\frac {\ln \left (c\,x^n\right )\,\left (b\,e\,x^2+\frac {b\,d}{3}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.64, size = 75, normalized size = 1.42 \[ - \frac {a d}{3 x^{3}} - \frac {a e}{x} - \frac {b d n \log {\relax (x )}}{3 x^{3}} - \frac {b d n}{9 x^{3}} - \frac {b d \log {\relax (c )}}{3 x^{3}} - \frac {b e n \log {\relax (x )}}{x} - \frac {b e n}{x} - \frac {b e \log {\relax (c )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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