Optimal. Leaf size=151 \[ \frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{f \left (a+b x^4\right )^4}{16 b}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15} \]
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Rubi [A] time = 0.108231, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1582, 1657} \[ \frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{f \left (a+b x^4\right )^4}{16 b}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1657
Rubi steps
\begin{align*} \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^3 \, dx &=\frac{f \left (a+b x^4\right )^4}{16 b}+\int \left (c+d x+e x^2\right ) \left (a+b x^4\right )^3 \, dx\\ &=\frac{f \left (a+b x^4\right )^4}{16 b}+\int \left (a^3 c+a^3 d x+a^3 e x^2+3 a^2 b c x^4+3 a^2 b d x^5+3 a^2 b e x^6+3 a b^2 c x^8+3 a b^2 d x^9+3 a b^2 e x^{10}+b^3 c x^{12}+b^3 d x^{13}+b^3 e x^{14}\right ) \, dx\\ &=a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\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}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15}+\frac{f \left (a+b x^4\right )^4}{16 b}\\ \end{align*}
Mathematica [A] time = 0.0049483, size = 180, normalized size = 1.19 \[ \frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a^2 b f x^8+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{1}{4} a^3 f x^4+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{4} a b^2 f x^{12}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15}+\frac{1}{16} b^3 f x^{16} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 151, normalized size = 1. \begin{align*}{\frac{{b}^{3}f{x}^{16}}{16}}+{\frac{{b}^{3}e{x}^{15}}{15}}+{\frac{{b}^{3}d{x}^{14}}{14}}+{\frac{{b}^{3}c{x}^{13}}{13}}+{\frac{a{b}^{2}f{x}^{12}}{4}}+{\frac{3\,a{b}^{2}e{x}^{11}}{11}}+{\frac{3\,a{b}^{2}d{x}^{10}}{10}}+{\frac{a{b}^{2}c{x}^{9}}{3}}+{\frac{3\,fb{a}^{2}{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{f{a}^{3}{x}^{4}}{4}}+{\frac{{a}^{3}e{x}^{3}}{3}}+{\frac{{a}^{3}d{x}^{2}}{2}}+{a}^{3}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.910649, size = 203, normalized size = 1.34 \begin{align*} \frac{1}{16} \, b^{3} f x^{16} + \frac{1}{15} \, b^{3} e x^{15} + \frac{1}{14} \, b^{3} d x^{14} + \frac{1}{13} \, b^{3} c x^{13} + \frac{1}{4} \, a b^{2} f x^{12} + \frac{3}{11} \, a b^{2} e x^{11} + \frac{3}{10} \, a b^{2} d x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{3}{8} \, a^{2} b f 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} f x^{4} + \frac{1}{3} \, a^{3} e x^{3} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38173, size = 375, normalized size = 2.48 \begin{align*} \frac{1}{16} x^{16} f b^{3} + \frac{1}{15} x^{15} e b^{3} + \frac{1}{14} x^{14} d b^{3} + \frac{1}{13} x^{13} c b^{3} + \frac{1}{4} x^{12} f b^{2} a + \frac{3}{11} x^{11} e b^{2} a + \frac{3}{10} x^{10} d b^{2} a + \frac{1}{3} x^{9} c b^{2} a + \frac{3}{8} x^{8} f b a^{2} + \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} f a^{3} + \frac{1}{3} x^{3} e a^{3} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087819, size = 180, normalized size = 1.19 \begin{align*} a^{3} c x + \frac{a^{3} d x^{2}}{2} + \frac{a^{3} e x^{3}}{3} + \frac{a^{3} f 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^{2} b f x^{8}}{8} + \frac{a b^{2} c x^{9}}{3} + \frac{3 a b^{2} d x^{10}}{10} + \frac{3 a b^{2} e x^{11}}{11} + \frac{a b^{2} f x^{12}}{4} + \frac{b^{3} c x^{13}}{13} + \frac{b^{3} d x^{14}}{14} + \frac{b^{3} e x^{15}}{15} + \frac{b^{3} f x^{16}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06258, size = 208, normalized size = 1.38 \begin{align*} \frac{1}{16} \, b^{3} f x^{16} + \frac{1}{15} \, b^{3} x^{15} e + \frac{1}{14} \, b^{3} d x^{14} + \frac{1}{13} \, b^{3} c x^{13} + \frac{1}{4} \, a b^{2} f x^{12} + \frac{3}{11} \, a b^{2} x^{11} e + \frac{3}{10} \, a b^{2} d x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{3}{8} \, a^{2} b f 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} f x^{4} + \frac{1}{3} \, a^{3} x^{3} e + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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