Optimal. Leaf size=27 \[ -\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2304} \[ -\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx &=-\frac {b n}{4 x^2}-\frac {a+b \log \left (c x^n\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.19 \[ -\frac {a}{2 x^2}-\frac {b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 23, normalized size = 0.85 \[ -\frac {2 \, b n \log \relax (x) + b n + 2 \, b \log \relax (c) + 2 \, a}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 27, normalized size = 1.00 \[ -\frac {b n \log \relax (x)}{2 \, x^{2}} - \frac {b n + 2 \, b \log \relax (c) + 2 \, a}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 111, normalized size = 4.11 \[ -\frac {b \ln \left (x^{n}\right )}{2 x^{2}}-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b n +2 b \ln \relax (c )+2 a}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 26, normalized size = 0.96 \[ -\frac {b n}{4 \, x^{2}} - \frac {b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 26, normalized size = 0.96 \[ -\frac {\frac {a}{2}+\frac {b\,n}{4}}{x^2}-\frac {b\,\ln \left (c\,x^n\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 37, normalized size = 1.37 \[ - \frac {a}{2 x^{2}} - \frac {b n \log {\relax (x )}}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\relax (c )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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