Optimal. Leaf size=166 \[ \frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12} \]
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Rubi [A] time = 0.124775, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1628} \[ \frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx &=\int \left (4 a^3 b c+\frac{a^4 c}{x^3}+\frac{a^4 d}{x^2}+\frac{a^4 e}{x}+4 a^3 b d x+4 a^3 b e x^2+6 a^2 b^2 c x^3+6 a^2 b^2 d x^4+6 a^2 b^2 e x^5+4 a b^3 c x^6+4 a b^3 d x^7+4 a b^3 e x^8+b^4 c x^9+b^4 d x^{10}+b^4 e x^{11}\right ) \, dx\\ &=-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3+\frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12}+a^4 e \log (x)\\ \end{align*}
Mathematica [A] time = 0.0089017, size = 166, normalized size = 1. \[ \frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 147, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}c}{2\,{x}^{2}}}-{\frac{{a}^{4}d}{x}}+4\,{a}^{3}bcx+2\,{a}^{3}bd{x}^{2}+{\frac{4\,{a}^{3}be{x}^{3}}{3}}+{\frac{3\,{a}^{2}{b}^{2}c{x}^{4}}{2}}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{5}}{5}}+{a}^{2}{b}^{2}e{x}^{6}+{\frac{4\,a{b}^{3}c{x}^{7}}{7}}+{\frac{a{b}^{3}d{x}^{8}}{2}}+{\frac{4\,a{b}^{3}e{x}^{9}}{9}}+{\frac{{b}^{4}c{x}^{10}}{10}}+{\frac{{b}^{4}d{x}^{11}}{11}}+{\frac{{b}^{4}e{x}^{12}}{12}}+{a}^{4}e\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949211, size = 197, normalized size = 1.19 \begin{align*} \frac{1}{12} \, b^{4} e x^{12} + \frac{1}{11} \, b^{4} d x^{11} + \frac{1}{10} \, b^{4} c x^{10} + \frac{4}{9} \, a b^{3} e x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} e x^{6} + \frac{6}{5} \, a^{2} b^{2} d x^{5} + \frac{3}{2} \, a^{2} b^{2} c x^{4} + \frac{4}{3} \, a^{3} b e x^{3} + 2 \, a^{3} b d x^{2} + 4 \, a^{3} b c x + a^{4} e \log \left (x\right ) - \frac{2 \, a^{4} d x + a^{4} c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52702, size = 413, normalized size = 2.49 \begin{align*} \frac{1155 \, b^{4} e x^{14} + 1260 \, b^{4} d x^{13} + 1386 \, b^{4} c x^{12} + 6160 \, a b^{3} e x^{11} + 6930 \, a b^{3} d x^{10} + 7920 \, a b^{3} c x^{9} + 13860 \, a^{2} b^{2} e x^{8} + 16632 \, a^{2} b^{2} d x^{7} + 20790 \, a^{2} b^{2} c x^{6} + 18480 \, a^{3} b e x^{5} + 27720 \, a^{3} b d x^{4} + 55440 \, a^{3} b c x^{3} + 13860 \, a^{4} e x^{2} \log \left (x\right ) - 13860 \, a^{4} d x - 6930 \, a^{4} c}{13860 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.558303, size = 173, normalized size = 1.04 \begin{align*} a^{4} e \log{\left (x \right )} + 4 a^{3} b c x + 2 a^{3} b d x^{2} + \frac{4 a^{3} b e x^{3}}{3} + \frac{3 a^{2} b^{2} c x^{4}}{2} + \frac{6 a^{2} b^{2} d x^{5}}{5} + a^{2} b^{2} e x^{6} + \frac{4 a b^{3} c x^{7}}{7} + \frac{a b^{3} d x^{8}}{2} + \frac{4 a b^{3} e x^{9}}{9} + \frac{b^{4} c x^{10}}{10} + \frac{b^{4} d x^{11}}{11} + \frac{b^{4} e x^{12}}{12} - \frac{a^{4} c + 2 a^{4} d x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05512, size = 205, normalized size = 1.23 \begin{align*} \frac{1}{12} \, b^{4} x^{12} e + \frac{1}{11} \, b^{4} d x^{11} + \frac{1}{10} \, b^{4} c x^{10} + \frac{4}{9} \, a b^{3} x^{9} e + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} x^{6} e + \frac{6}{5} \, a^{2} b^{2} d x^{5} + \frac{3}{2} \, a^{2} b^{2} c x^{4} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + 4 \, a^{3} b c x + a^{4} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{4} d x + a^{4} c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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