Optimal. Leaf size=53 \[ \frac{d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{r}-\frac{b e n x^r}{r^2} \]
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Rubi [A] time = 0.0880529, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {14, 2351, 2301, 2304} \[ \frac{d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{r}-\frac{b e n x^r}{r^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2351
Rule 2301
Rule 2304
Rubi steps
\begin{align*} \int \frac{\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\int \left (\frac{d \left (a+b \log \left (c x^n\right )\right )}{x}+e x^{-1+r} \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ &=d \int \frac{a+b \log \left (c x^n\right )}{x} \, dx+e \int x^{-1+r} \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{b e n x^r}{r^2}+\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}\\ \end{align*}
Mathematica [A] time = 0.0877378, size = 54, normalized size = 1.02 \[ \frac{e x^r (a r-b n)}{r^2}+a d \log (x)+\frac{b d \log ^2\left (c x^n\right )}{2 n}+\frac{b e x^r \log \left (c x^n\right )}{r} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.175, size = 278, normalized size = 5.3 \begin{align*}{\frac{b \left ( dr\ln \left ( x \right ) +e{x}^{r} \right ) \ln \left ({x}^{n} \right ) }{r}}+{\frac{i}{2}}\pi \,\ln \left ( x \right ) bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{2}}\pi \,\ln \left ( x \right ) bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{2}}\pi \,\ln \left ( x \right ) bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{2}}\pi \,\ln \left ( x \right ) bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{{\frac{i}{2}}\pi \,be{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{x}^{r}}{r}}-{\frac{{\frac{i}{2}}\pi \,be{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){x}^{r}}{r}}-{\frac{{\frac{i}{2}}\pi \,be \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}{x}^{r}}{r}}+{\frac{{\frac{i}{2}}\pi \,be \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ){x}^{r}}{r}}-{\frac{bdn \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( x \right ) \ln \left ( c \right ) bd+{\frac{\ln \left ( c \right ) be{x}^{r}}{r}}+\ln \left ( x \right ) ad+{\frac{{x}^{r}ae}{r}}-{\frac{{x}^{r}ben}{{r}^{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 = 1.32992, size = 167, normalized size = 3.15 \begin{align*} \frac{b d n r^{2} \log \left (x\right )^{2} + 2 \,{\left (b e n r \log \left (x\right ) + b e r \log \left (c\right ) - b e n + a e r\right )} x^{r} + 2 \,{\left (b d r^{2} \log \left (c\right ) + a d r^{2}\right )} \log \left (x\right )}{2 \, r^{2}} \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 [A] time = 1.30248, size = 93, normalized size = 1.75 \begin{align*} \frac{1}{2} \, b d n \log \left (x\right )^{2} + \frac{b n x^{r} e \log \left (x\right )}{r} + b d \log \left (c\right ) \log \left (x\right ) + \frac{b x^{r} e \log \left (c\right )}{r} + a d \log \left (x\right ) - \frac{b n x^{r} e}{r^{2}} + \frac{a x^{r} e}{r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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