Optimal. Leaf size=74 \[ \frac{1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{2}{25} b d e n x^5-\frac{1}{36} b e^2 n x^6 \]
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Rubi [A] time = 0.0897376, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {43, 2334, 12, 14} \[ \frac{1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{2}{25} b d e n x^5-\frac{1}{36} b e^2 n x^6 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rule 14
Rubi steps
\begin{align*} \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{60} x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right ) \, dx\\ &=\frac{1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right ) \, dx\\ &=\frac{1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \left (15 d^2 x^3+24 d e x^4+10 e^2 x^5\right ) \, dx\\ &=-\frac{1}{16} b d^2 n x^4-\frac{2}{25} b d e n x^5-\frac{1}{36} b e^2 n x^6+\frac{1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0507269, size = 81, normalized size = 1.09 \[ \frac{x^4 \left (60 a \left (15 d^2+24 d e x+10 e^2 x^2\right )+60 b \left (15 d^2+24 d e x+10 e^2 x^2\right ) \log \left (c x^n\right )-b n \left (225 d^2+288 d e x+100 e^2 x^2\right )\right )}{3600} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.245, size = 432, normalized size = 5.8 \begin{align*}{\frac{b{x}^{4} \left ( 10\,{e}^{2}{x}^{2}+24\,dex+15\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{60}}-{\frac{i}{5}}\pi \,bde{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{5}}\pi \,bde{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{6}}{6}}-{\frac{b{e}^{2}n{x}^{6}}{36}}+{\frac{a{e}^{2}{x}^{6}}{6}}+{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{5}}\pi \,bde{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{2\,\ln \left ( c \right ) bde{x}^{5}}{5}}-{\frac{2\,bden{x}^{5}}{25}}+{\frac{2\,ade{x}^{5}}{5}}-{\frac{i}{5}}\pi \,bde{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,b{e}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{8}}\pi \,b{d}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) b{d}^{2}{x}^{4}}{4}}-{\frac{b{d}^{2}n{x}^{4}}{16}}+{\frac{a{d}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17073, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{36} \, b e^{2} n x^{6} + \frac{1}{6} \, b e^{2} x^{6} \log \left (c x^{n}\right ) - \frac{2}{25} \, b d e n x^{5} + \frac{1}{6} \, a e^{2} x^{6} + \frac{2}{5} \, b d e x^{5} \log \left (c x^{n}\right ) - \frac{1}{16} \, b d^{2} n x^{4} + \frac{2}{5} \, a d e x^{5} + \frac{1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01816, size = 293, normalized size = 3.96 \begin{align*} -\frac{1}{36} \,{\left (b e^{2} n - 6 \, a e^{2}\right )} x^{6} - \frac{2}{25} \,{\left (b d e n - 5 \, a d e\right )} x^{5} - \frac{1}{16} \,{\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} + \frac{1}{60} \,{\left (10 \, b e^{2} x^{6} + 24 \, b d e x^{5} + 15 \, b d^{2} x^{4}\right )} \log \left (c\right ) + \frac{1}{60} \,{\left (10 \, b e^{2} n x^{6} + 24 \, b d e n x^{5} + 15 \, b d^{2} n x^{4}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 14.3758, size = 158, normalized size = 2.14 \begin{align*} \frac{a d^{2} x^{4}}{4} + \frac{2 a d e x^{5}}{5} + \frac{a e^{2} x^{6}}{6} + \frac{b d^{2} n x^{4} \log{\left (x \right )}}{4} - \frac{b d^{2} n x^{4}}{16} + \frac{b d^{2} x^{4} \log{\left (c \right )}}{4} + \frac{2 b d e n x^{5} \log{\left (x \right )}}{5} - \frac{2 b d e n x^{5}}{25} + \frac{2 b d e x^{5} \log{\left (c \right )}}{5} + \frac{b e^{2} n x^{6} \log{\left (x \right )}}{6} - \frac{b e^{2} n x^{6}}{36} + \frac{b e^{2} x^{6} \log{\left (c \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33211, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{6} \, b n x^{6} e^{2} \log \left (x\right ) + \frac{2}{5} \, b d n x^{5} e \log \left (x\right ) - \frac{1}{36} \, b n x^{6} e^{2} - \frac{2}{25} \, b d n x^{5} e + \frac{1}{6} \, b x^{6} e^{2} \log \left (c\right ) + \frac{2}{5} \, b d x^{5} e \log \left (c\right ) + \frac{1}{4} \, b d^{2} n x^{4} \log \left (x\right ) - \frac{1}{16} \, b d^{2} n x^{4} + \frac{1}{6} \, a x^{6} e^{2} + \frac{2}{5} \, a d x^{5} e + \frac{1}{4} \, b d^{2} x^{4} \log \left (c\right ) + \frac{1}{4} \, a d^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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