Optimal. Leaf size=46 \[ \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1}}{d (m+1)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2304} \[ \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1}}{d (m+1)^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {b n (d x)^{1+m}}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.70 \[ \frac {x (d x)^m \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )}{(m+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 52, normalized size = 1.13 \[ \frac {{\left ({\left (b m + b\right )} n x \log \relax (x) + {\left (b m + b\right )} x \log \relax (c) + {\left (a m - b n + a\right )} x\right )} e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{m^{2} + 2 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 95, normalized size = 2.07 \[ \frac {b d^{m} m n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {b d^{m} n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {b d^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (d x\right )^{m} b x \log \relax (c)}{m + 1} + \frac {\left (d x\right )^{m} a x}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 371, normalized size = 8.07 \[ \frac {b x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) m}{2}} \ln \left (x^{n}\right )}{m +1}-\frac {\left (i \pi b m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b m \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b m \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b m \ln \relax (c )-2 a m +2 b n -2 b \ln \relax (c )-2 a \right ) x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) m}{2}}}{2 \left (m +1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 57, normalized size = 1.24 \[ -\frac {b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (d x\right )^{m + 1} b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.45, size = 192, normalized size = 4.17 \[ \begin {cases} \frac {a d^{m} m x x^{m}}{m^{2} + 2 m + 1} + \frac {a d^{m} x x^{m}}{m^{2} + 2 m + 1} + \frac {b d^{m} m n x x^{m} \log {\relax (x )}}{m^{2} + 2 m + 1} + \frac {b d^{m} m x x^{m} \log {\relax (c )}}{m^{2} + 2 m + 1} + \frac {b d^{m} n x x^{m} \log {\relax (x )}}{m^{2} + 2 m + 1} - \frac {b d^{m} n x x^{m}}{m^{2} + 2 m + 1} + \frac {b d^{m} x x^{m} \log {\relax (c )}}{m^{2} + 2 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} a \log {\relax (x )} & \text {for}\: b = 0 \\- \left (- a - b \log {\relax (c )}\right ) \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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