Optimal. Leaf size=74 \[ \frac{1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d^2 n x^3-\frac{1}{8} b d e n x^4-\frac{1}{25} b e^2 n x^5 \]
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Rubi [A] time = 0.079526, 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} \[ \frac{1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d^2 n x^3-\frac{1}{8} b d e n x^4-\frac{1}{25} b e^2 n x^5 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rule 14
Rubi steps
\begin{align*} \int x^2 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{30} x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right ) \, dx\\ &=\frac{1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{30} (b n) \int x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right ) \, dx\\ &=\frac{1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{30} (b n) \int \left (10 d^2 x^2+15 d e x^3+6 e^2 x^4\right ) \, dx\\ &=-\frac{1}{9} b d^2 n x^3-\frac{1}{8} b d e n x^4-\frac{1}{25} b e^2 n x^5+\frac{1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.039512, size = 81, normalized size = 1.09 \[ \frac{x^3 \left (60 a \left (10 d^2+15 d e x+6 e^2 x^2\right )+60 b \left (10 d^2+15 d e x+6 e^2 x^2\right ) \log \left (c x^n\right )-b n \left (200 d^2+225 d e x+72 e^2 x^2\right )\right )}{1800} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.21, size = 432, normalized size = 5.8 \begin{align*}{\frac{b{x}^{3} \left ( 6\,{e}^{2}{x}^{2}+15\,dex+10\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{30}}-{\frac{i}{6}}\pi \,b{d}^{2}{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 \,b{d}^{2}{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{6}}\pi \,b{d}^{2}{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{10}}\pi \,b{e}^{2}{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}^{5}}{5}}-{\frac{b{e}^{2}n{x}^{5}}{25}}+{\frac{a{e}^{2}{x}^{5}}{5}}+{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{4}}\pi \,bde{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{6}}\pi \,b{d}^{2}{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) bde{x}^{4}}{2}}-{\frac{bden{x}^{4}}{8}}+{\frac{ade{x}^{4}}{2}}-{\frac{i}{4}}\pi \,bde{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{4}}\pi \,bde{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{4}}\pi \,bde{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}^{3}}{3}}-{\frac{b{d}^{2}n{x}^{3}}{9}}+{\frac{a{d}^{2}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22446, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{25} \, b e^{2} n x^{5} + \frac{1}{5} \, b e^{2} x^{5} \log \left (c x^{n}\right ) - \frac{1}{8} \, b d e n x^{4} + \frac{1}{5} \, a e^{2} x^{5} + \frac{1}{2} \, b d e x^{4} \log \left (c x^{n}\right ) - \frac{1}{9} \, b d^{2} n x^{3} + \frac{1}{2} \, a d e x^{4} + \frac{1}{3} \, b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.993726, size = 288, normalized size = 3.89 \begin{align*} -\frac{1}{25} \,{\left (b e^{2} n - 5 \, a e^{2}\right )} x^{5} - \frac{1}{8} \,{\left (b d e n - 4 \, a d e\right )} x^{4} - \frac{1}{9} \,{\left (b d^{2} n - 3 \, a d^{2}\right )} x^{3} + \frac{1}{30} \,{\left (6 \, b e^{2} x^{5} + 15 \, b d e x^{4} + 10 \, b d^{2} x^{3}\right )} \log \left (c\right ) + \frac{1}{30} \,{\left (6 \, b e^{2} n x^{5} + 15 \, b d e n x^{4} + 10 \, b d^{2} 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 = 21.0525, size = 151, normalized size = 2.04 \begin{align*} \frac{a d^{2} x^{3}}{3} + \frac{a d e x^{4}}{2} + \frac{a e^{2} x^{5}}{5} + \frac{b d^{2} n x^{3} \log{\left (x \right )}}{3} - \frac{b d^{2} n x^{3}}{9} + \frac{b d^{2} x^{3} \log{\left (c \right )}}{3} + \frac{b d e n x^{4} \log{\left (x \right )}}{2} - \frac{b d e n x^{4}}{8} + \frac{b d e x^{4} \log{\left (c \right )}}{2} + \frac{b e^{2} n x^{5} \log{\left (x \right )}}{5} - \frac{b e^{2} n x^{5}}{25} + \frac{b e^{2} x^{5} \log{\left (c \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33896, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{5} \, b n x^{5} e^{2} \log \left (x\right ) + \frac{1}{2} \, b d n x^{4} e \log \left (x\right ) - \frac{1}{25} \, b n x^{5} e^{2} - \frac{1}{8} \, b d n x^{4} e + \frac{1}{5} \, b x^{5} e^{2} \log \left (c\right ) + \frac{1}{2} \, b d x^{4} e \log \left (c\right ) + \frac{1}{3} \, b d^{2} n x^{3} \log \left (x\right ) - \frac{1}{9} \, b d^{2} n x^{3} + \frac{1}{5} \, a x^{5} e^{2} + \frac{1}{2} \, a d x^{4} e + \frac{1}{3} \, b d^{2} x^{3} \log \left (c\right ) + \frac{1}{3} \, a d^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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