Optimal. Leaf size=38 \[ \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}-\frac {b n (f x)^m}{f m^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2304} \[ \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}-\frac {b n (f x)^m}{f m^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {b n (f x)^m}{f m^2}+\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.76 \[ \frac {(f x)^m \left (a m+b m \log \left (c x^n\right )-b n\right )}{f m^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 42, normalized size = 1.11 \[ \frac {{\left (b m n x \log \relax (x) + b m x \log \relax (c) + {\left (a m - b n\right )} x\right )} e^{\left ({\left (m - 1\right )} \log \relax (f) + {\left (m - 1\right )} \log \relax (x)\right )}}{m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 80, normalized size = 2.11 \[ \frac {b \frac {1}{f}^{m} x^{m} {\left | f \right |}^{2 \, m} \log \relax (c)}{f m} + \frac {b f^{m} n x^{m} \log \relax (x)}{f m} + \frac {a \frac {1}{f}^{m} x^{m} {\left | f \right |}^{2 \, m}}{f m} - \frac {b f^{m} n x^{m}}{f m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 281, normalized size = 7.39 \[ \frac {b x \,{\mathrm e}^{\frac {\left (m -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )+i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )^{2}-i \pi \mathrm {csgn}\left (i f x \right )^{3}+2 \ln \relax (f )+2 \ln \relax (x )\right )}{2}} \ln \left (x^{n}\right )}{m}+\frac {\left (-i \pi b m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b m \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b m \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b m \ln \relax (c )+2 a m -2 b n \right ) x \,{\mathrm e}^{\frac {\left (m -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )+i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )^{2}-i \pi \mathrm {csgn}\left (i f x \right )^{3}+2 \ln \relax (f )+2 \ln \relax (x )\right )}{2}}}{2 m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 48, normalized size = 1.26 \[ -\frac {b f^{m - 1} n x^{m}}{m^{2}} + \frac {\left (f x\right )^{m} b \log \left (c x^{n}\right )}{f m} + \frac {\left (f x\right )^{m} a}{f m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (f\,x\right )}^{m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.25, size = 148, normalized size = 3.89 \[ \begin {cases} \tilde {\infty } \left (a x + b n x \log {\relax (x )} - b n x + b x \log {\relax (c )}\right ) & \text {for}\: f = 0 \wedge m = 0 \\\frac {\begin {cases} a \log {\relax (x )} & \text {for}\: b = 0 \\- \left (- a - b \log {\relax (c )}\right ) \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}}{f} & \text {for}\: m = 0 \\0^{m - 1} \left (a x + b n x \log {\relax (x )} - b n x + b x \log {\relax (c )}\right ) & \text {for}\: f = 0 \\\frac {a f^{m} x^{m}}{f m} + \frac {b f^{m} n x^{m} \log {\relax (x )}}{f m} + \frac {b f^{m} x^{m} \log {\relax (c )}}{f m} - \frac {b f^{m} n x^{m}}{f m^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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