Optimal. Leaf size=74 \[ \frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{1}{18} b d e n x^6-\frac{1}{64} b e^2 n x^8 \]
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Rubi [A] time = 0.088268, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 12, 14} \[ \frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d^2 n x^4-\frac{1}{18} b d e n x^6-\frac{1}{64} b e^2 n x^8 \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2334
Rule 12
Rule 14
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{24} x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \, dx\\ &=\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{24} (b n) \int x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \, dx\\ &=\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{24} (b n) \int \left (6 d^2 x^3+8 d e x^5+3 e^2 x^7\right ) \, dx\\ &=-\frac{1}{16} b d^2 n x^4-\frac{1}{18} b d e n x^6-\frac{1}{64} b e^2 n x^8+\frac{1}{24} \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0365365, size = 87, normalized size = 1.18 \[ \frac{1}{576} x^4 \left (24 a \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+24 b \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \log \left (c x^n\right )-b n \left (36 d^2+32 d e x^2+9 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.198, size = 434, normalized size = 5.9 \begin{align*}{\frac{b{x}^{4} \left ( 3\,{e}^{2}{x}^{4}+8\,de{x}^{2}+6\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{24}}-{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8}{\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}{\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}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,bde{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{8}}{8}}-{\frac{b{e}^{2}n{x}^{8}}{64}}+{\frac{a{e}^{2}{x}^{8}}{8}}+{\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}{6}}\pi \,bde{x}^{6}{\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 ) ^{3}+{\frac{i}{6}}\pi \,bde{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) bde{x}^{6}}{3}}-{\frac{bden{x}^{6}}{18}}+{\frac{ade{x}^{6}}{3}}+{\frac{i}{6}}\pi \,bde{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{16}}\pi \,b{e}^{2}{x}^{8}{\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 = 1.12407, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{64} \, b e^{2} n x^{8} + \frac{1}{8} \, b e^{2} x^{8} \log \left (c x^{n}\right ) + \frac{1}{8} \, a e^{2} x^{8} - \frac{1}{18} \, b d e n x^{6} + \frac{1}{3} \, b d e x^{6} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d e x^{6} - \frac{1}{16} \, b d^{2} n x^{4} + \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 = 1.34008, size = 285, normalized size = 3.85 \begin{align*} -\frac{1}{64} \,{\left (b e^{2} n - 8 \, a e^{2}\right )} x^{8} - \frac{1}{18} \,{\left (b d e n - 6 \, a d e\right )} x^{6} - \frac{1}{16} \,{\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} + \frac{1}{24} \,{\left (3 \, b e^{2} x^{8} + 8 \, b d e x^{6} + 6 \, b d^{2} x^{4}\right )} \log \left (c\right ) + \frac{1}{24} \,{\left (3 \, b e^{2} n x^{8} + 8 \, b d e n x^{6} + 6 \, 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 = 13.8342, size = 151, normalized size = 2.04 \begin{align*} \frac{a d^{2} x^{4}}{4} + \frac{a d e x^{6}}{3} + \frac{a e^{2} x^{8}}{8} + \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{b d e n x^{6} \log{\left (x \right )}}{3} - \frac{b d e n x^{6}}{18} + \frac{b d e x^{6} \log{\left (c \right )}}{3} + \frac{b e^{2} n x^{8} \log{\left (x \right )}}{8} - \frac{b e^{2} n x^{8}}{64} + \frac{b e^{2} x^{8} \log{\left (c \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37819, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{8} \, b n x^{8} e^{2} \log \left (x\right ) - \frac{1}{64} \, b n x^{8} e^{2} + \frac{1}{8} \, b x^{8} e^{2} \log \left (c\right ) + \frac{1}{3} \, b d n x^{6} e \log \left (x\right ) + \frac{1}{8} \, a x^{8} e^{2} - \frac{1}{18} \, b d n x^{6} e + \frac{1}{3} \, b d x^{6} e \log \left (c\right ) + \frac{1}{3} \, a d x^{6} e + \frac{1}{4} \, b d^{2} n x^{4} \log \left (x\right ) - \frac{1}{16} \, b d^{2} n x^{4} + \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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