Optimal. Leaf size=48 \[ \frac{1}{6} \left (3 d x^2+2 e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d n x^2-\frac{1}{9} b e n x^3 \]
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Rubi [A] time = 0.0361271, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {43, 2334, 12} \[ \frac{1}{6} \left (3 d x^2+2 e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d n x^2-\frac{1}{9} b e n x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rubi steps
\begin{align*} \int x (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{6} \left (3 d x^2+2 e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{6} x (3 d+2 e x) \, dx\\ &=\frac{1}{6} \left (3 d x^2+2 e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int x (3 d+2 e x) \, dx\\ &=\frac{1}{6} \left (3 d x^2+2 e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \left (3 d x+2 e x^2\right ) \, dx\\ &=-\frac{1}{4} b d n x^2-\frac{1}{9} b e n x^3+\frac{1}{6} \left (3 d x^2+2 e x^3\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0193183, size = 48, normalized size = 1. \[ \frac{1}{36} x^2 \left (6 a (3 d+2 e x)+6 b (3 d+2 e x) \log \left (c x^n\right )-b n (9 d+4 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.207, size = 264, normalized size = 5.5 \begin{align*}{\frac{b{x}^{2} \left ( 2\,ex+3\,d \right ) \ln \left ({x}^{n} \right ) }{6}}+{\frac{i}{6}}\pi \,be{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,be{x}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,be{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{6}}\pi \,be{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{3}}{3}}-{\frac{ben{x}^{3}}{9}}+{\frac{ae{x}^{3}}{3}}+{\frac{i}{4}}\pi \,bd{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{4}}\pi \,bd{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{4}}\pi \,bd{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{4}}\pi \,bd{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{2}}{2}}-{\frac{bdn{x}^{2}}{4}}+{\frac{ad{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18804, size = 77, normalized size = 1.6 \begin{align*} -\frac{1}{9} \, b e n x^{3} + \frac{1}{3} \, b e x^{3} \log \left (c x^{n}\right ) - \frac{1}{4} \, b d n x^{2} + \frac{1}{3} \, a e x^{3} + \frac{1}{2} \, b d x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a d x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00222, size = 176, normalized size = 3.67 \begin{align*} -\frac{1}{9} \,{\left (b e n - 3 \, a e\right )} x^{3} - \frac{1}{4} \,{\left (b d n - 2 \, a d\right )} x^{2} + \frac{1}{6} \,{\left (2 \, b e x^{3} + 3 \, b d x^{2}\right )} \log \left (c\right ) + \frac{1}{6} \,{\left (2 \, b e n x^{3} + 3 \, b d n x^{2}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.01117, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{2}}{2} + \frac{a e x^{3}}{3} + \frac{b d n x^{2} \log{\left (x \right )}}{2} - \frac{b d n x^{2}}{4} + \frac{b d x^{2} \log{\left (c \right )}}{2} + \frac{b e n x^{3} \log{\left (x \right )}}{3} - \frac{b e n x^{3}}{9} + \frac{b e x^{3} \log{\left (c \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22798, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{3} \, b n x^{3} e \log \left (x\right ) - \frac{1}{9} \, b n x^{3} e + \frac{1}{3} \, b x^{3} e \log \left (c\right ) + \frac{1}{2} \, b d n x^{2} \log \left (x\right ) - \frac{1}{4} \, b d n x^{2} + \frac{1}{3} \, a x^{3} e + \frac{1}{2} \, b d x^{2} \log \left (c\right ) + \frac{1}{2} \, a d x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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