Optimal. Leaf size=156 \[ \frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{c \left (a+b x^4\right )^4}{16 b}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19} \]
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Rubi [A] time = 0.112745, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1582, 1850} \[ \frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{c \left (a+b x^4\right )^4}{16 b}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1850
Rubi steps
\begin{align*} \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^3 \, dx &=\frac{c \left (a+b x^4\right )^4}{16 b}+\int \left (a+b x^4\right )^3 \left (-c x^3+x^3 \left (c+d x+e x^2+f x^3\right )\right ) \, dx\\ &=\frac{c \left (a+b x^4\right )^4}{16 b}+\int \left (a^3 d x^4+a^3 e x^5+a^3 f x^6+3 a^2 b d x^8+3 a^2 b e x^9+3 a^2 b f x^{10}+3 a b^2 d x^{12}+3 a b^2 e x^{13}+3 a b^2 f x^{14}+b^3 d x^{16}+b^3 e x^{17}+b^3 f x^{18}\right ) \, dx\\ &=\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19}+\frac{c \left (a+b x^4\right )^4}{16 b}\\ \end{align*}
Mathematica [A] time = 0.0052431, size = 185, normalized size = 1.19 \[ \frac{3}{8} a^2 b c x^8+\frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{1}{4} a b^2 c x^{12}+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{1}{16} b^3 c x^{16}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 154, normalized size = 1. \begin{align*}{\frac{{b}^{3}f{x}^{19}}{19}}+{\frac{{b}^{3}e{x}^{18}}{18}}+{\frac{{b}^{3}d{x}^{17}}{17}}+{\frac{{b}^{3}c{x}^{16}}{16}}+{\frac{a{b}^{2}f{x}^{15}}{5}}+{\frac{3\,a{b}^{2}e{x}^{14}}{14}}+{\frac{3\,a{b}^{2}d{x}^{13}}{13}}+{\frac{ac{b}^{2}{x}^{12}}{4}}+{\frac{3\,{a}^{2}bf{x}^{11}}{11}}+{\frac{3\,{a}^{2}be{x}^{10}}{10}}+{\frac{{a}^{2}bd{x}^{9}}{3}}+{\frac{3\,b{a}^{2}c{x}^{8}}{8}}+{\frac{{a}^{3}f{x}^{7}}{7}}+{\frac{{a}^{3}e{x}^{6}}{6}}+{\frac{{a}^{3}d{x}^{5}}{5}}+{\frac{{a}^{3}c{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.916298, size = 207, normalized size = 1.33 \begin{align*} \frac{1}{19} \, b^{3} f x^{19} + \frac{1}{18} \, b^{3} e x^{18} + \frac{1}{17} \, b^{3} d x^{17} + \frac{1}{16} \, b^{3} c x^{16} + \frac{1}{5} \, a b^{2} f x^{15} + \frac{3}{14} \, a b^{2} e x^{14} + \frac{3}{13} \, a b^{2} d x^{13} + \frac{1}{4} \, a b^{2} c x^{12} + \frac{3}{11} \, a^{2} b f x^{11} + \frac{3}{10} \, a^{2} b e x^{10} + \frac{1}{3} \, a^{2} b d x^{9} + \frac{3}{8} \, a^{2} b c x^{8} + \frac{1}{7} \, a^{3} f x^{7} + \frac{1}{6} \, a^{3} e x^{6} + \frac{1}{5} \, a^{3} d x^{5} + \frac{1}{4} \, a^{3} c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44069, size = 390, normalized size = 2.5 \begin{align*} \frac{1}{19} x^{19} f b^{3} + \frac{1}{18} x^{18} e b^{3} + \frac{1}{17} x^{17} d b^{3} + \frac{1}{16} x^{16} c b^{3} + \frac{1}{5} x^{15} f b^{2} a + \frac{3}{14} x^{14} e b^{2} a + \frac{3}{13} x^{13} d b^{2} a + \frac{1}{4} x^{12} c b^{2} a + \frac{3}{11} x^{11} f b a^{2} + \frac{3}{10} x^{10} e b a^{2} + \frac{1}{3} x^{9} d b a^{2} + \frac{3}{8} x^{8} c b a^{2} + \frac{1}{7} x^{7} f a^{3} + \frac{1}{6} x^{6} e a^{3} + \frac{1}{5} x^{5} d a^{3} + \frac{1}{4} x^{4} c a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.084306, size = 184, normalized size = 1.18 \begin{align*} \frac{a^{3} c x^{4}}{4} + \frac{a^{3} d x^{5}}{5} + \frac{a^{3} e x^{6}}{6} + \frac{a^{3} f x^{7}}{7} + \frac{3 a^{2} b c x^{8}}{8} + \frac{a^{2} b d x^{9}}{3} + \frac{3 a^{2} b e x^{10}}{10} + \frac{3 a^{2} b f x^{11}}{11} + \frac{a b^{2} c x^{12}}{4} + \frac{3 a b^{2} d x^{13}}{13} + \frac{3 a b^{2} e x^{14}}{14} + \frac{a b^{2} f x^{15}}{5} + \frac{b^{3} c x^{16}}{16} + \frac{b^{3} d x^{17}}{17} + \frac{b^{3} e x^{18}}{18} + \frac{b^{3} f x^{19}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07847, size = 212, normalized size = 1.36 \begin{align*} \frac{1}{19} \, b^{3} f x^{19} + \frac{1}{18} \, b^{3} x^{18} e + \frac{1}{17} \, b^{3} d x^{17} + \frac{1}{16} \, b^{3} c x^{16} + \frac{1}{5} \, a b^{2} f x^{15} + \frac{3}{14} \, a b^{2} x^{14} e + \frac{3}{13} \, a b^{2} d x^{13} + \frac{1}{4} \, a b^{2} c x^{12} + \frac{3}{11} \, a^{2} b f x^{11} + \frac{3}{10} \, a^{2} b x^{10} e + \frac{1}{3} \, a^{2} b d x^{9} + \frac{3}{8} \, a^{2} b c x^{8} + \frac{1}{7} \, a^{3} f x^{7} + \frac{1}{6} \, a^{3} x^{6} e + \frac{1}{5} \, a^{3} d x^{5} + \frac{1}{4} \, a^{3} c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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