Optimal. Leaf size=125 \[ \frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10} \]
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Rubi [A] time = 0.0922203, antiderivative size = 125, 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} \[ \frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10} \]
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 )^3}{x^2} \, dx &=\int \left (a^3 e+\frac{a^3 c}{x^2}+\frac{a^3 d}{x}+3 a^2 b c x+3 a^2 b d x^2+3 a^2 b e x^3+3 a b^2 c x^4+3 a b^2 d x^5+3 a b^2 e x^6+b^3 c x^7+b^3 d x^8+b^3 e x^9\right ) \, dx\\ &=-\frac{a^3 c}{x}+a^3 e x+\frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10}+a^3 d \log (x)\\ \end{align*}
Mathematica [A] time = 0.0083931, size = 125, normalized size = 1. \[ \frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}c}{x}}+{a}^{3}ex+{\frac{3\,{a}^{2}bc{x}^{2}}{2}}+{a}^{2}bd{x}^{3}+{\frac{3\,{a}^{2}be{x}^{4}}{4}}+{\frac{3\,a{b}^{2}c{x}^{5}}{5}}+{\frac{a{b}^{2}d{x}^{6}}{2}}+{\frac{3\,a{b}^{2}e{x}^{7}}{7}}+{\frac{{b}^{3}c{x}^{8}}{8}}+{\frac{{b}^{3}d{x}^{9}}{9}}+{\frac{{b}^{3}e{x}^{10}}{10}}+{a}^{3}d\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940401, size = 147, normalized size = 1.18 \begin{align*} \frac{1}{10} \, b^{3} e x^{10} + \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{8} \, b^{3} c x^{8} + \frac{3}{7} \, a b^{2} e x^{7} + \frac{1}{2} \, a b^{2} d x^{6} + \frac{3}{5} \, a b^{2} c x^{5} + \frac{3}{4} \, a^{2} b e x^{4} + a^{2} b d x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + a^{3} e x + a^{3} d \log \left (x\right ) - \frac{a^{3} c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45317, size = 305, normalized size = 2.44 \begin{align*} \frac{252 \, b^{3} e x^{11} + 280 \, b^{3} d x^{10} + 315 \, b^{3} c x^{9} + 1080 \, a b^{2} e x^{8} + 1260 \, a b^{2} d x^{7} + 1512 \, a b^{2} c x^{6} + 1890 \, a^{2} b e x^{5} + 2520 \, a^{2} b d x^{4} + 3780 \, a^{2} b c x^{3} + 2520 \, a^{3} e x^{2} + 2520 \, a^{3} d x \log \left (x\right ) - 2520 \, a^{3} c}{2520 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.39922, size = 128, normalized size = 1.02 \begin{align*} - \frac{a^{3} c}{x} + a^{3} d \log{\left (x \right )} + a^{3} e x + \frac{3 a^{2} b c x^{2}}{2} + a^{2} b d x^{3} + \frac{3 a^{2} b e x^{4}}{4} + \frac{3 a b^{2} c x^{5}}{5} + \frac{a b^{2} d x^{6}}{2} + \frac{3 a b^{2} e x^{7}}{7} + \frac{b^{3} c x^{8}}{8} + \frac{b^{3} d x^{9}}{9} + \frac{b^{3} e x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05804, size = 154, normalized size = 1.23 \begin{align*} \frac{1}{10} \, b^{3} x^{10} e + \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{8} \, b^{3} c x^{8} + \frac{3}{7} \, a b^{2} x^{7} e + \frac{1}{2} \, a b^{2} d x^{6} + \frac{3}{5} \, a b^{2} c x^{5} + \frac{3}{4} \, a^{2} b x^{4} e + a^{2} b d x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + a^{3} x e + a^{3} d \log \left ({\left | x \right |}\right ) - \frac{a^{3} c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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