Optimal. Leaf size=152 \[ \frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}-\frac{3 b d^2 e n x^r}{r^2}-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3 b d e^2 n x^{2 r}}{4 r^2}-\frac{b e^3 n x^{3 r}}{9 r^2} \]
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Rubi [A] time = 0.170992, antiderivative size = 124, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {266, 43, 2334, 12, 14, 2301} \[ \frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+6 d^3 \log (x)+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^r}{r^2}-\frac{1}{2} b d^3 n \log ^2(x)-\frac{3 b d e^2 n x^{2 r}}{4 r^2}-\frac{b e^3 n x^{3 r}}{9 r^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int \frac{\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{e x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )+6 d^3 r \log (x)}{6 r x} \, dx\\ &=\frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b n) \int \frac{e x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )+6 d^3 r \log (x)}{x} \, dx}{6 r}\\ &=\frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b n) \int \left (18 d^2 e x^{-1+r}+9 d e^2 x^{-1+2 r}+2 e^3 x^{-1+3 r}+\frac{6 d^3 r \log (x)}{x}\right ) \, dx}{6 r}\\ &=-\frac{3 b d^2 e n x^r}{r^2}-\frac{3 b d e^2 n x^{2 r}}{4 r^2}-\frac{b e^3 n x^{3 r}}{9 r^2}+\frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^3 n\right ) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{3 b d^2 e n x^r}{r^2}-\frac{3 b d e^2 n x^{2 r}}{4 r^2}-\frac{b e^3 n x^{3 r}}{9 r^2}-\frac{1}{2} b d^3 n \log ^2(x)+\frac{1}{6} \left (\frac{18 d^2 e x^r}{r}+\frac{9 d e^2 x^{2 r}}{r}+\frac{2 e^3 x^{3 r}}{r}+6 d^3 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.353775, size = 132, normalized size = 0.87 \[ \frac{1}{36} \left (\frac{e x^r \left (6 a r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )-b n \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )\right )}{r^2}+\frac{6 b e x^r \log \left (c x^n\right ) \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )}{r}+\frac{18 b d^3 \log ^2\left (c x^n\right )}{n}\right )+a d^3 \log (x) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.16, size = 693, normalized size = 4.6 \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 [A] time = 1.39656, size = 419, normalized size = 2.76 \begin{align*} \frac{18 \, b d^{3} n r^{2} \log \left (x\right )^{2} + 4 \,{\left (3 \, b e^{3} n r \log \left (x\right ) + 3 \, b e^{3} r \log \left (c\right ) - b e^{3} n + 3 \, a e^{3} r\right )} x^{3 \, r} + 27 \,{\left (2 \, b d e^{2} n r \log \left (x\right ) + 2 \, b d e^{2} r \log \left (c\right ) - b d e^{2} n + 2 \, a d e^{2} r\right )} x^{2 \, r} + 108 \,{\left (b d^{2} e n r \log \left (x\right ) + b d^{2} e r \log \left (c\right ) - b d^{2} e n + a d^{2} e r\right )} x^{r} + 36 \,{\left (b d^{3} r^{2} \log \left (c\right ) + a d^{3} r^{2}\right )} \log \left (x\right )}{36 \, 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 = 1.35322, size = 284, normalized size = 1.87 \begin{align*} \frac{1}{2} \, b d^{3} n \log \left (x\right )^{2} + \frac{3 \, b d^{2} n x^{r} e \log \left (x\right )}{r} + b d^{3} \log \left (c\right ) \log \left (x\right ) + \frac{3 \, b d^{2} x^{r} e \log \left (c\right )}{r} + a d^{3} \log \left (x\right ) + \frac{3 \, b d n x^{2 \, r} e^{2} \log \left (x\right )}{2 \, r} - \frac{3 \, b d^{2} n x^{r} e}{r^{2}} + \frac{3 \, a d^{2} x^{r} e}{r} + \frac{3 \, b d x^{2 \, r} e^{2} \log \left (c\right )}{2 \, r} + \frac{b n x^{3 \, r} e^{3} \log \left (x\right )}{3 \, r} - \frac{3 \, b d n x^{2 \, r} e^{2}}{4 \, r^{2}} + \frac{3 \, a d x^{2 \, r} e^{2}}{2 \, r} + \frac{b x^{3 \, r} e^{3} \log \left (c\right )}{3 \, r} - \frac{b n x^{3 \, r} e^{3}}{9 \, r^{2}} + \frac{a x^{3 \, r} e^{3}}{3 \, r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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