Optimal. Leaf size=138 \[ \frac{3}{4} a^2 b^2 c x^8+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{5} a^3 b c x^5+\frac{4}{7} a^3 b e x^7+\frac{1}{2} a^4 c x^2+\frac{1}{4} a^4 e x^4+\frac{4}{11} a b^3 c x^{11}+\frac{4}{13} a b^3 e x^{13}+\frac{d \left (a+b x^3\right )^5}{15 b}+\frac{1}{14} b^4 c x^{14}+\frac{1}{16} b^4 e x^{16} \]
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Rubi [A] time = 0.0934948, antiderivative size = 138, 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} \[ \frac{3}{4} a^2 b^2 c x^8+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{5} a^3 b c x^5+\frac{4}{7} a^3 b e x^7+\frac{1}{2} a^4 c x^2+\frac{1}{4} a^4 e x^4+\frac{4}{11} a b^3 c x^{11}+\frac{4}{13} a b^3 e x^{13}+\frac{d \left (a+b x^3\right )^5}{15 b}+\frac{1}{14} b^4 c x^{14}+\frac{1}{16} b^4 e x^{16} \]
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 )^4 \, dx &=\frac{d \left (a+b x^3\right )^5}{15 b}+\int \left (a+b x^3\right )^4 \left (-d x^2+x \left (c+d x+e x^2\right )\right ) \, dx\\ &=\frac{d \left (a+b x^3\right )^5}{15 b}+\int \left (a^4 c x+a^4 e x^3+4 a^3 b c x^4+4 a^3 b e x^6+6 a^2 b^2 c x^7+6 a^2 b^2 e x^9+4 a b^3 c x^{10}+4 a b^3 e x^{12}+b^4 c x^{13}+b^4 e x^{15}\right ) \, dx\\ &=\frac{1}{2} a^4 c x^2+\frac{1}{4} a^4 e x^4+\frac{4}{5} a^3 b c x^5+\frac{4}{7} a^3 b e x^7+\frac{3}{4} a^2 b^2 c x^8+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{11} a b^3 c x^{11}+\frac{4}{13} a b^3 e x^{13}+\frac{1}{14} b^4 c x^{14}+\frac{1}{16} b^4 e x^{16}+\frac{d \left (a+b x^3\right )^5}{15 b}\\ \end{align*}
Mathematica [A] time = 0.0042877, size = 181, normalized size = 1.31 \[ \frac{3}{4} a^2 b^2 c x^8+\frac{2}{3} a^2 b^2 d x^9+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{1}{2} a^4 c x^2+\frac{1}{3} a^4 d x^3+\frac{1}{4} a^4 e x^4+\frac{4}{11} a b^3 c x^{11}+\frac{1}{3} a b^3 d x^{12}+\frac{4}{13} a b^3 e x^{13}+\frac{1}{14} b^4 c x^{14}+\frac{1}{15} b^4 d x^{15}+\frac{1}{16} b^4 e x^{16} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 152, normalized size = 1.1 \begin{align*}{\frac{{b}^{4}e{x}^{16}}{16}}+{\frac{{b}^{4}d{x}^{15}}{15}}+{\frac{{b}^{4}c{x}^{14}}{14}}+{\frac{4\,a{b}^{3}e{x}^{13}}{13}}+{\frac{a{b}^{3}d{x}^{12}}{3}}+{\frac{4\,a{b}^{3}c{x}^{11}}{11}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{10}}{5}}+{\frac{2\,{a}^{2}{b}^{2}d{x}^{9}}{3}}+{\frac{3\,{a}^{2}{b}^{2}c{x}^{8}}{4}}+{\frac{4\,{a}^{3}be{x}^{7}}{7}}+{\frac{2\,{a}^{3}bd{x}^{6}}{3}}+{\frac{4\,{a}^{3}bc{x}^{5}}{5}}+{\frac{{a}^{4}e{x}^{4}}{4}}+{\frac{{a}^{4}d{x}^{3}}{3}}+{\frac{{a}^{4}c{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939755, size = 204, normalized size = 1.48 \begin{align*} \frac{1}{16} \, b^{4} e x^{16} + \frac{1}{15} \, b^{4} d x^{15} + \frac{1}{14} \, b^{4} c x^{14} + \frac{4}{13} \, a b^{3} e x^{13} + \frac{1}{3} \, a b^{3} d x^{12} + \frac{4}{11} \, a b^{3} c x^{11} + \frac{3}{5} \, a^{2} b^{2} e x^{10} + \frac{2}{3} \, a^{2} b^{2} d x^{9} + \frac{3}{4} \, a^{2} b^{2} c x^{8} + \frac{4}{7} \, a^{3} b e x^{7} + \frac{2}{3} \, a^{3} b d x^{6} + \frac{4}{5} \, a^{3} b c x^{5} + \frac{1}{4} \, a^{4} e x^{4} + \frac{1}{3} \, a^{4} d x^{3} + \frac{1}{2} \, a^{4} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30116, size = 371, normalized size = 2.69 \begin{align*} \frac{1}{16} x^{16} e b^{4} + \frac{1}{15} x^{15} d b^{4} + \frac{1}{14} x^{14} c b^{4} + \frac{4}{13} x^{13} e b^{3} a + \frac{1}{3} x^{12} d b^{3} a + \frac{4}{11} x^{11} c b^{3} a + \frac{3}{5} x^{10} e b^{2} a^{2} + \frac{2}{3} x^{9} d b^{2} a^{2} + \frac{3}{4} x^{8} c b^{2} a^{2} + \frac{4}{7} x^{7} e b a^{3} + \frac{2}{3} x^{6} d b a^{3} + \frac{4}{5} x^{5} c b a^{3} + \frac{1}{4} x^{4} e a^{4} + \frac{1}{3} x^{3} d a^{4} + \frac{1}{2} x^{2} c a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087054, size = 185, normalized size = 1.34 \begin{align*} \frac{a^{4} c x^{2}}{2} + \frac{a^{4} d x^{3}}{3} + \frac{a^{4} e x^{4}}{4} + \frac{4 a^{3} b c x^{5}}{5} + \frac{2 a^{3} b d x^{6}}{3} + \frac{4 a^{3} b e x^{7}}{7} + \frac{3 a^{2} b^{2} c x^{8}}{4} + \frac{2 a^{2} b^{2} d x^{9}}{3} + \frac{3 a^{2} b^{2} e x^{10}}{5} + \frac{4 a b^{3} c x^{11}}{11} + \frac{a b^{3} d x^{12}}{3} + \frac{4 a b^{3} e x^{13}}{13} + \frac{b^{4} c x^{14}}{14} + \frac{b^{4} d x^{15}}{15} + \frac{b^{4} e x^{16}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05148, size = 211, normalized size = 1.53 \begin{align*} \frac{1}{16} \, b^{4} x^{16} e + \frac{1}{15} \, b^{4} d x^{15} + \frac{1}{14} \, b^{4} c x^{14} + \frac{4}{13} \, a b^{3} x^{13} e + \frac{1}{3} \, a b^{3} d x^{12} + \frac{4}{11} \, a b^{3} c x^{11} + \frac{3}{5} \, a^{2} b^{2} x^{10} e + \frac{2}{3} \, a^{2} b^{2} d x^{9} + \frac{3}{4} \, a^{2} b^{2} c x^{8} + \frac{4}{7} \, a^{3} b x^{7} e + \frac{2}{3} \, a^{3} b d x^{6} + \frac{4}{5} \, a^{3} b c x^{5} + \frac{1}{4} \, a^{4} x^{4} e + \frac{1}{3} \, a^{4} d x^{3} + \frac{1}{2} \, a^{4} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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