Optimal. Leaf size=110 \[ \frac{3}{5} a^2 b c x^5+\frac{3}{7} a^2 b e x^7+\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{3}{8} a b^2 c x^8+\frac{3}{10} a b^2 e x^{10}+\frac{d \left (a+b x^3\right )^4}{12 b}+\frac{1}{11} b^3 c x^{11}+\frac{1}{13} b^3 e x^{13} \]
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Rubi [A] time = 0.0693723, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1582, 1850} \[ \frac{3}{5} a^2 b c x^5+\frac{3}{7} a^2 b e x^7+\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{3}{8} a b^2 c x^8+\frac{3}{10} a b^2 e x^{10}+\frac{d \left (a+b x^3\right )^4}{12 b}+\frac{1}{11} b^3 c x^{11}+\frac{1}{13} b^3 e x^{13} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1850
Rubi steps
\begin{align*} \int x \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx &=\frac{d \left (a+b x^3\right )^4}{12 b}+\int \left (a+b x^3\right )^3 \left (-d x^2+x \left (c+d x+e x^2\right )\right ) \, dx\\ &=\frac{d \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 c x+a^3 e x^3+3 a^2 b c x^4+3 a^2 b e x^6+3 a b^2 c x^7+3 a b^2 e x^9+b^3 c x^{10}+b^3 e x^{12}\right ) \, dx\\ &=\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{3}{5} a^2 b c x^5+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a b^2 c x^8+\frac{3}{10} a b^2 e x^{10}+\frac{1}{11} b^3 c x^{11}+\frac{1}{13} b^3 e x^{13}+\frac{d \left (a+b x^3\right )^4}{12 b}\\ \end{align*}
Mathematica [A] time = 0.0036847, size = 139, normalized size = 1.26 \[ \frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{3}{8} a b^2 c x^8+\frac{1}{3} a b^2 d x^9+\frac{3}{10} a b^2 e x^{10}+\frac{1}{11} b^3 c x^{11}+\frac{1}{12} b^3 d x^{12}+\frac{1}{13} b^3 e x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 116, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}e{x}^{13}}{13}}+{\frac{{b}^{3}d{x}^{12}}{12}}+{\frac{{b}^{3}c{x}^{11}}{11}}+{\frac{3\,a{b}^{2}e{x}^{10}}{10}}+{\frac{a{b}^{2}d{x}^{9}}{3}}+{\frac{3\,a{b}^{2}c{x}^{8}}{8}}+{\frac{3\,{a}^{2}be{x}^{7}}{7}}+{\frac{{a}^{2}bd{x}^{6}}{2}}+{\frac{3\,{a}^{2}bc{x}^{5}}{5}}+{\frac{{a}^{3}e{x}^{4}}{4}}+{\frac{{a}^{3}d{x}^{3}}{3}}+{\frac{{a}^{3}c{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.919483, size = 155, normalized size = 1.41 \begin{align*} \frac{1}{13} \, b^{3} e x^{13} + \frac{1}{12} \, b^{3} d x^{12} + \frac{1}{11} \, b^{3} c x^{11} + \frac{3}{10} \, a b^{2} e x^{10} + \frac{1}{3} \, a b^{2} d x^{9} + \frac{3}{8} \, a b^{2} c x^{8} + \frac{3}{7} \, a^{2} b e x^{7} + \frac{1}{2} \, a^{2} b d x^{6} + \frac{3}{5} \, a^{2} b c x^{5} + \frac{1}{4} \, a^{3} e x^{4} + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{2} \, a^{3} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26776, size = 285, normalized size = 2.59 \begin{align*} \frac{1}{13} x^{13} e b^{3} + \frac{1}{12} x^{12} d b^{3} + \frac{1}{11} x^{11} c b^{3} + \frac{3}{10} x^{10} e b^{2} a + \frac{1}{3} x^{9} d b^{2} a + \frac{3}{8} x^{8} c b^{2} a + \frac{3}{7} x^{7} e b a^{2} + \frac{1}{2} x^{6} d b a^{2} + \frac{3}{5} x^{5} c b a^{2} + \frac{1}{4} x^{4} e a^{3} + \frac{1}{3} x^{3} d a^{3} + \frac{1}{2} x^{2} c a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076247, size = 138, normalized size = 1.25 \begin{align*} \frac{a^{3} c x^{2}}{2} + \frac{a^{3} d x^{3}}{3} + \frac{a^{3} e x^{4}}{4} + \frac{3 a^{2} b c x^{5}}{5} + \frac{a^{2} b d x^{6}}{2} + \frac{3 a^{2} b e x^{7}}{7} + \frac{3 a b^{2} c x^{8}}{8} + \frac{a b^{2} d x^{9}}{3} + \frac{3 a b^{2} e x^{10}}{10} + \frac{b^{3} c x^{11}}{11} + \frac{b^{3} d x^{12}}{12} + \frac{b^{3} e x^{13}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08196, size = 161, normalized size = 1.46 \begin{align*} \frac{1}{13} \, b^{3} x^{13} e + \frac{1}{12} \, b^{3} d x^{12} + \frac{1}{11} \, b^{3} c x^{11} + \frac{3}{10} \, a b^{2} x^{10} e + \frac{1}{3} \, a b^{2} d x^{9} + \frac{3}{8} \, a b^{2} c x^{8} + \frac{3}{7} \, a^{2} b x^{7} e + \frac{1}{2} \, a^{2} b d x^{6} + \frac{3}{5} \, a^{2} b c x^{5} + \frac{1}{4} \, a^{3} x^{4} e + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{2} \, a^{3} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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