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 = 0.0151003, antiderivative size = 46, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0117765, size = 32, normalized size = 0.7 \[ \frac{x (d x)^m \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )}{(m+1)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.103, size = 371, normalized size = 8.1 \begin{align*}{\frac{bx\ln \left ({x}^{n} \right ) }{1+m}{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}}-{\frac{ \left ( -i\pi \,b{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}m+i\pi \,b{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) m+i\pi \,b \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}m-i\pi \,b \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) m-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,b\ln \left ( c \right ) m-2\,b\ln \left ( c \right ) -2\,am+2\,bn-2\,a \right ) x}{2\, \left ( 1+m \right ) ^{2}}{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03528, size = 142, normalized size = 3.09 \begin{align*} \frac{{\left ({\left (b m + b\right )} n x \log \left (x\right ) +{\left (b m + b\right )} x \log \left (c\right ) +{\left (a m - b n + a\right )} x\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{2} + 2 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28532, size = 128, normalized size = 2.78 \begin{align*} \frac{b d^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{b d^{m} n x x^{m} \log \left (x\right )}{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 \left (c\right )}{m + 1} + \frac{\left (d x\right )^{m} a x}{m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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