Optimal. Leaf size=48 \[ d x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )-b d n x-\frac{1}{9} b e n x^3 \]
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Rubi [A] time = 0.0176313, antiderivative size = 41, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2313} \[ \frac{1}{3} \left (3 d x+e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-b d n x-\frac{1}{9} b e n x^3 \]
Antiderivative was successfully verified.
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Rule 2313
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{3} \left (3 d x+e x^3\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d+\frac{e x^2}{3}\right ) \, dx\\ &=-b d n x-\frac{1}{9} b e n x^3+\frac{1}{3} \left (3 d x+e x^3\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0014115, size = 55, normalized size = 1.15 \[ a d x+\frac{1}{3} a e x^3+b d x \log \left (c x^n\right )+\frac{1}{3} b e x^3 \log \left (c x^n\right )-b d n x-\frac{1}{9} b e n x^3 \]
Antiderivative was successfully verified.
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Maple [C] time = 0.187, size = 247, normalized size = 5.2 \begin{align*}{\frac{bx \left ( e{x}^{2}+3\,d \right ) \ln \left ({x}^{n} \right ) }{3}}+{\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{i}{2}}\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) x-{\frac{i}{2}}\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}x+{\frac{i}{2}}\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x+{\frac{\ln \left ( c \right ) be{x}^{3}}{3}}-{\frac{ben{x}^{3}}{9}}+{\frac{ae{x}^{3}}{3}}+\ln \left ( c \right ) bdx-bdnx+axd \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04568, size = 66, normalized size = 1.38 \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}{3} \, a e x^{3} - b d n x + b d x \log \left (c x^{n}\right ) + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23191, size = 154, normalized size = 3.21 \begin{align*} -\frac{1}{9} \,{\left (b e n - 3 \, a e\right )} x^{3} -{\left (b d n - a d\right )} x + \frac{1}{3} \,{\left (b e x^{3} + 3 \, b d x\right )} \log \left (c\right ) + \frac{1}{3} \,{\left (b e n x^{3} + 3 \, b d n x\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14463, size = 73, normalized size = 1.52 \begin{align*} a d x + \frac{a e x^{3}}{3} + b d n x \log{\left (x \right )} - b d n x + b d x \log{\left (c \right )} + \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 = 1.34424, size = 84, normalized size = 1.75 \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}{3} \, a x^{3} e + b d n x \log \left (x\right ) - b d n x + b d x \log \left (c\right ) + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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