Optimal. Leaf size=67 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac{b d n}{x}-\frac{b e n x^{r-1}}{(1-r)^2} \]
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Rubi [A] time = 0.0766622, antiderivative size = 58, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {14, 2334, 12} \[ -\left (\frac{d}{x}+\frac{e x^{r-1}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{x}-\frac{b e n x^{r-1}}{(1-r)^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2334
Rule 12
Rubi steps
\begin{align*} \int \frac{\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\left (\frac{d}{x}+\frac{e x^{-1+r}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d+d r-e x^r}{(1-r) x^2} \, dx\\ &=-\left (\frac{d}{x}+\frac{e x^{-1+r}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b n) \int \frac{-d+d r-e x^r}{x^2} \, dx}{1-r}\\ &=-\left (\frac{d}{x}+\frac{e x^{-1+r}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b n) \int \left (\frac{d (-1+r)}{x^2}-e x^{-2+r}\right ) \, dx}{1-r}\\ &=-\frac{b d n}{x}-\frac{b e n x^{-1+r}}{(1-r)^2}-\left (\frac{d}{x}+\frac{e x^{-1+r}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.103402, size = 67, normalized size = 1. \[ -\frac{a (r-1) \left (d (r-1)-e x^r\right )+b (r-1) \log \left (c x^n\right ) \left (d (r-1)-e x^r\right )+b n \left (d (r-1)^2+e x^r\right )}{(r-1)^2 x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.152, size = 614, normalized size = 9.2 \begin{align*} \text{result too large to display} \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 [B] time = 1.28411, size = 311, normalized size = 4.64 \begin{align*} -\frac{b d n +{\left (b d n + a d\right )} r^{2} + a d - 2 \,{\left (b d n + a d\right )} r +{\left (b e n - a e r + a e -{\left (b e r - b e\right )} \log \left (c\right ) -{\left (b e n r - b e n\right )} \log \left (x\right )\right )} x^{r} +{\left (b d r^{2} - 2 \, b d r + b d\right )} \log \left (c\right ) +{\left (b d n r^{2} - 2 \, b d n r + b d n\right )} \log \left (x\right )}{{\left (r^{2} - 2 \, r + 1\right )} x} \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 = 1.32393, size = 261, normalized size = 3.9 \begin{align*} \frac{b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 2 \, r + 1\right )} x} + \frac{b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac{b d n \log \left (x\right )}{x} - \frac{b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac{b d n}{x} - \frac{b n x^{r} e}{{\left (r^{2} - 2 \, r + 1\right )} x} + \frac{a r x^{r} e}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac{b d \log \left (c\right )}{x} - \frac{b x^{r} e \log \left (c\right )}{{\left (r^{2} - 2 \, r + 1\right )} x} - \frac{a d}{x} - \frac{a x^{r} e}{{\left (r^{2} - 2 \, r + 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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