Optimal. Leaf size=126 \[ 3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9 \]
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Rubi [A] time = 0.0864589, antiderivative size = 126, 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} \[ 3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9 \]
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^3} \, dx &=\int \left (3 a^2 b c+\frac{a^3 c}{x^3}+\frac{a^3 d}{x^2}+\frac{a^3 e}{x}+3 a^2 b d x+3 a^2 b e x^2+3 a b^2 c x^3+3 a b^2 d x^4+3 a b^2 e x^5+b^3 c x^6+b^3 d x^7+b^3 e x^8\right ) \, dx\\ &=-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9+a^3 e \log (x)\\ \end{align*}
Mathematica [A] time = 0.0090364, size = 126, normalized size = 1. \[ 3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 111, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}c}{2\,{x}^{2}}}-{\frac{{a}^{3}d}{x}}+3\,{a}^{2}bcx+{\frac{3\,{a}^{2}bd{x}^{2}}{2}}+{a}^{2}be{x}^{3}+{\frac{3\,a{b}^{2}c{x}^{4}}{4}}+{\frac{3\,a{b}^{2}d{x}^{5}}{5}}+{\frac{a{b}^{2}e{x}^{6}}{2}}+{\frac{{b}^{3}c{x}^{7}}{7}}+{\frac{{b}^{3}d{x}^{8}}{8}}+{\frac{{b}^{3}e{x}^{9}}{9}}+{a}^{3}e\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935902, size = 149, normalized size = 1.18 \begin{align*} \frac{1}{9} \, b^{3} e x^{9} + \frac{1}{8} \, b^{3} d x^{8} + \frac{1}{7} \, b^{3} c x^{7} + \frac{1}{2} \, a b^{2} e x^{6} + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{4} \, a b^{2} c x^{4} + a^{2} b e x^{3} + \frac{3}{2} \, a^{2} b d x^{2} + 3 \, a^{2} b c x + a^{3} e \log \left (x\right ) - \frac{2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46691, size = 308, normalized size = 2.44 \begin{align*} \frac{280 \, b^{3} e x^{11} + 315 \, b^{3} d x^{10} + 360 \, b^{3} c x^{9} + 1260 \, a b^{2} e x^{8} + 1512 \, a b^{2} d x^{7} + 1890 \, a b^{2} c x^{6} + 2520 \, a^{2} b e x^{5} + 3780 \, a^{2} b d x^{4} + 7560 \, a^{2} b c x^{3} + 2520 \, a^{3} e x^{2} \log \left (x\right ) - 2520 \, a^{3} d x - 1260 \, a^{3} c}{2520 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.460926, size = 129, normalized size = 1.02 \begin{align*} a^{3} e \log{\left (x \right )} + 3 a^{2} b c x + \frac{3 a^{2} b d x^{2}}{2} + a^{2} b e x^{3} + \frac{3 a b^{2} c x^{4}}{4} + \frac{3 a b^{2} d x^{5}}{5} + \frac{a b^{2} e x^{6}}{2} + \frac{b^{3} c x^{7}}{7} + \frac{b^{3} d x^{8}}{8} + \frac{b^{3} e x^{9}}{9} - \frac{a^{3} c + 2 a^{3} d x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0553, size = 155, normalized size = 1.23 \begin{align*} \frac{1}{9} \, b^{3} x^{9} e + \frac{1}{8} \, b^{3} d x^{8} + \frac{1}{7} \, b^{3} c x^{7} + \frac{1}{2} \, a b^{2} x^{6} e + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{4} \, a b^{2} c x^{4} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + 3 \, a^{2} b c x + a^{3} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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