Optimal. Leaf size=198 \[ \frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left (a+b x^4\right )^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]
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Rubi [A] time = 0.150432, antiderivative size = 198, 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} \[ \frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left (a+b x^4\right )^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]
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 )^4 \, dx &=\frac{c \left (a+b x^4\right )^5}{20 b}+\int \left (a+b x^4\right )^4 \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 )^5}{20 b}+\int \left (a^4 d x^4+a^4 e x^5+a^4 f x^6+4 a^3 b d x^8+4 a^3 b e x^9+4 a^3 b f x^{10}+6 a^2 b^2 d x^{12}+6 a^2 b^2 e x^{13}+6 a^2 b^2 f x^{14}+4 a b^3 d x^{16}+4 a b^3 e x^{17}+4 a b^3 f x^{18}+b^4 d x^{20}+b^4 e x^{21}+b^4 f x^{22}\right ) \, dx\\ &=\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23}+\frac{c \left (a+b x^4\right )^5}{20 b}\\ \end{align*}
Mathematica [A] time = 0.0062771, size = 241, normalized size = 1.22 \[ \frac{1}{2} a^2 b^2 c x^{12}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{1}{2} a^3 b c x^8+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{4} a^4 c x^4+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{1}{4} a b^3 c x^{16}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{1}{20} b^4 c x^{20}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 202, normalized size = 1. \begin{align*}{\frac{{b}^{4}f{x}^{23}}{23}}+{\frac{{b}^{4}e{x}^{22}}{22}}+{\frac{{b}^{4}d{x}^{21}}{21}}+{\frac{c{b}^{4}{x}^{20}}{20}}+{\frac{4\,a{b}^{3}f{x}^{19}}{19}}+{\frac{2\,a{b}^{3}e{x}^{18}}{9}}+{\frac{4\,a{b}^{3}d{x}^{17}}{17}}+{\frac{ac{b}^{3}{x}^{16}}{4}}+{\frac{2\,{a}^{2}{b}^{2}f{x}^{15}}{5}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{14}}{7}}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{13}}{13}}+{\frac{{a}^{2}{b}^{2}c{x}^{12}}{2}}+{\frac{4\,{a}^{3}bf{x}^{11}}{11}}+{\frac{2\,{a}^{3}be{x}^{10}}{5}}+{\frac{4\,{a}^{3}bd{x}^{9}}{9}}+{\frac{c{a}^{3}b{x}^{8}}{2}}+{\frac{{a}^{4}f{x}^{7}}{7}}+{\frac{{a}^{4}e{x}^{6}}{6}}+{\frac{{a}^{4}d{x}^{5}}{5}}+{\frac{{a}^{4}c{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.920182, size = 271, normalized size = 1.37 \begin{align*} \frac{1}{23} \, b^{4} f x^{23} + \frac{1}{22} \, b^{4} e x^{22} + \frac{1}{21} \, b^{4} d x^{21} + \frac{1}{20} \, b^{4} c x^{20} + \frac{4}{19} \, a b^{3} f x^{19} + \frac{2}{9} \, a b^{3} e x^{18} + \frac{4}{17} \, a b^{3} d x^{17} + \frac{1}{4} \, a b^{3} c x^{16} + \frac{2}{5} \, a^{2} b^{2} f x^{15} + \frac{3}{7} \, a^{2} b^{2} e x^{14} + \frac{6}{13} \, a^{2} b^{2} d x^{13} + \frac{1}{2} \, a^{2} b^{2} c x^{12} + \frac{4}{11} \, a^{3} b f x^{11} + \frac{2}{5} \, a^{3} b e x^{10} + \frac{4}{9} \, a^{3} b d x^{9} + \frac{1}{2} \, a^{3} b c x^{8} + \frac{1}{7} \, a^{4} f x^{7} + \frac{1}{6} \, a^{4} e x^{6} + \frac{1}{5} \, a^{4} d x^{5} + \frac{1}{4} \, a^{4} c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49758, size = 504, normalized size = 2.55 \begin{align*} \frac{1}{23} x^{23} f b^{4} + \frac{1}{22} x^{22} e b^{4} + \frac{1}{21} x^{21} d b^{4} + \frac{1}{20} x^{20} c b^{4} + \frac{4}{19} x^{19} f b^{3} a + \frac{2}{9} x^{18} e b^{3} a + \frac{4}{17} x^{17} d b^{3} a + \frac{1}{4} x^{16} c b^{3} a + \frac{2}{5} x^{15} f b^{2} a^{2} + \frac{3}{7} x^{14} e b^{2} a^{2} + \frac{6}{13} x^{13} d b^{2} a^{2} + \frac{1}{2} x^{12} c b^{2} a^{2} + \frac{4}{11} x^{11} f b a^{3} + \frac{2}{5} x^{10} e b a^{3} + \frac{4}{9} x^{9} d b a^{3} + \frac{1}{2} x^{8} c b a^{3} + \frac{1}{7} x^{7} f a^{4} + \frac{1}{6} x^{6} e a^{4} + \frac{1}{5} x^{5} d a^{4} + \frac{1}{4} x^{4} c a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.094753, size = 245, normalized size = 1.24 \begin{align*} \frac{a^{4} c x^{4}}{4} + \frac{a^{4} d x^{5}}{5} + \frac{a^{4} e x^{6}}{6} + \frac{a^{4} f x^{7}}{7} + \frac{a^{3} b c x^{8}}{2} + \frac{4 a^{3} b d x^{9}}{9} + \frac{2 a^{3} b e x^{10}}{5} + \frac{4 a^{3} b f x^{11}}{11} + \frac{a^{2} b^{2} c x^{12}}{2} + \frac{6 a^{2} b^{2} d x^{13}}{13} + \frac{3 a^{2} b^{2} e x^{14}}{7} + \frac{2 a^{2} b^{2} f x^{15}}{5} + \frac{a b^{3} c x^{16}}{4} + \frac{4 a b^{3} d x^{17}}{17} + \frac{2 a b^{3} e x^{18}}{9} + \frac{4 a b^{3} f x^{19}}{19} + \frac{b^{4} c x^{20}}{20} + \frac{b^{4} d x^{21}}{21} + \frac{b^{4} e x^{22}}{22} + \frac{b^{4} f x^{23}}{23} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08794, size = 278, normalized size = 1.4 \begin{align*} \frac{1}{23} \, b^{4} f x^{23} + \frac{1}{22} \, b^{4} x^{22} e + \frac{1}{21} \, b^{4} d x^{21} + \frac{1}{20} \, b^{4} c x^{20} + \frac{4}{19} \, a b^{3} f x^{19} + \frac{2}{9} \, a b^{3} x^{18} e + \frac{4}{17} \, a b^{3} d x^{17} + \frac{1}{4} \, a b^{3} c x^{16} + \frac{2}{5} \, a^{2} b^{2} f x^{15} + \frac{3}{7} \, a^{2} b^{2} x^{14} e + \frac{6}{13} \, a^{2} b^{2} d x^{13} + \frac{1}{2} \, a^{2} b^{2} c x^{12} + \frac{4}{11} \, a^{3} b f x^{11} + \frac{2}{5} \, a^{3} b x^{10} e + \frac{4}{9} \, a^{3} b d x^{9} + \frac{1}{2} \, a^{3} b c x^{8} + \frac{1}{7} \, a^{4} f x^{7} + \frac{1}{6} \, a^{4} x^{6} e + \frac{1}{5} \, a^{4} d x^{5} + \frac{1}{4} \, a^{4} c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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