Optimal. Leaf size=48 \[ \frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d n x^3-\frac{1}{16} b e n x^4 \]
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Rubi [A] time = 0.0501997, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {43, 2334, 12} \[ \frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d n x^3-\frac{1}{16} b e n x^4 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rubi steps
\begin{align*} \int x^2 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{12} x^2 (4 d+3 e x) \, dx\\ &=\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{12} (b n) \int x^2 (4 d+3 e x) \, dx\\ &=\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{12} (b n) \int \left (4 d x^2+3 e x^3\right ) \, dx\\ &=-\frac{1}{9} b d n x^3-\frac{1}{16} b e n x^4+\frac{1}{12} \left (4 d x^3+3 e x^4\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0237083, size = 45, normalized size = 0.94 \[ \frac{1}{144} x^3 \left (48 a d+36 a e x+12 b (4 d+3 e x) \log \left (c x^n\right )-16 b d n-9 b e n x\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.234, size = 264, normalized size = 5.5 \begin{align*}{\frac{b{x}^{3} \left ( 3\,ex+4\,d \right ) \ln \left ({x}^{n} \right ) }{12}}+{\frac{i}{8}}\pi \,be{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{8}}\pi \,be{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,be{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{8}}\pi \,be{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{4}}{4}}-{\frac{ben{x}^{4}}{16}}+{\frac{ae{x}^{4}}{4}}+{\frac{i}{6}}\pi \,bd{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,bd{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 \,bd{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{6}}\pi \,bd{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{3}}{3}}-{\frac{bdn{x}^{3}}{9}}+{\frac{ad{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16551, size = 77, normalized size = 1.6 \begin{align*} -\frac{1}{16} \, b e n x^{4} + \frac{1}{4} \, b e x^{4} \log \left (c x^{n}\right ) - \frac{1}{9} \, b d n x^{3} + \frac{1}{4} \, a e x^{4} + \frac{1}{3} \, b d x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.957587, size = 180, normalized size = 3.75 \begin{align*} -\frac{1}{16} \,{\left (b e n - 4 \, a e\right )} x^{4} - \frac{1}{9} \,{\left (b d n - 3 \, a d\right )} x^{3} + \frac{1}{12} \,{\left (3 \, b e x^{4} + 4 \, b d x^{3}\right )} \log \left (c\right ) + \frac{1}{12} \,{\left (3 \, b e n x^{4} + 4 \, b d n x^{3}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.34002, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{3}}{3} + \frac{a e x^{4}}{4} + \frac{b d n x^{3} \log{\left (x \right )}}{3} - \frac{b d n x^{3}}{9} + \frac{b d x^{3} \log{\left (c \right )}}{3} + \frac{b e n x^{4} \log{\left (x \right )}}{4} - \frac{b e n x^{4}}{16} + \frac{b e x^{4} \log{\left (c \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25848, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{4} \, b n x^{4} e \log \left (x\right ) - \frac{1}{16} \, b n x^{4} e + \frac{1}{4} \, b x^{4} e \log \left (c\right ) + \frac{1}{3} \, b d n x^{3} \log \left (x\right ) - \frac{1}{9} \, b d n x^{3} + \frac{1}{4} \, a x^{4} e + \frac{1}{3} \, b d x^{3} \log \left (c\right ) + \frac{1}{3} \, a d x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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