Optimal. Leaf size=198 \[ \frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left (a+b x^4\right )^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]
[Out]
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Rubi [A] time = 0.495861, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left (a+b x^4\right )^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]
Antiderivative was successfully verified.
[In] Int[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]
[Out]
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Rubi in Sympy [A] time = 64.7355, size = 201, normalized size = 1.02 \[ \frac{a^{4} d x^{5}}{5} + \frac{a^{4} e x^{6}}{6} + \frac{a^{4} f x^{7}}{7} + \frac{4 a^{3} b d x^{9}}{9} + \frac{2 a^{3} b e x^{10}}{5} + \frac{4 a^{3} b f x^{11}}{11} + \frac{6 a^{2} b^{2} d x^{13}}{13} + \frac{3 a^{2} b^{2} e x^{14}}{7} + \frac{2 a^{2} b^{2} f x^{15}}{5} + \frac{4 a b^{3} d x^{17}}{17} + \frac{2 a b^{3} e x^{18}}{9} + \frac{4 a b^{3} f x^{19}}{19} + \frac{b^{4} d x^{21}}{21} + \frac{b^{4} e x^{22}}{22} + \frac{b^{4} f x^{23}}{23} + \frac{c \left (a + b x^{4}\right )^{5}}{20 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**4,x)
[Out]
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Mathematica [A] time = 0.0101972, size = 241, normalized size = 1.22 \[ \frac{1}{4} a^4 c x^4+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{1}{2} a^3 b c x^8+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{2} a^2 b^2 c x^{12}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{1}{4} a b^3 c x^{16}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{1}{20} b^4 c x^{20}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]
[Out]
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Maple [A] time = 0.003, size = 202, normalized size = 1. \[{\frac{{b}^{4}f{x}^{23}}{23}}+{\frac{{b}^{4}e{x}^{22}}{22}}+{\frac{{b}^{4}d{x}^{21}}{21}}+{\frac{{b}^{4}c{x}^{20}}{20}}+{\frac{4\,a{b}^{3}f{x}^{19}}{19}}+{\frac{2\,a{b}^{3}e{x}^{18}}{9}}+{\frac{4\,a{b}^{3}d{x}^{17}}{17}}+{\frac{ac{b}^{3}{x}^{16}}{4}}+{\frac{2\,{a}^{2}{b}^{2}f{x}^{15}}{5}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{14}}{7}}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{13}}{13}}+{\frac{{a}^{2}{b}^{2}c{x}^{12}}{2}}+{\frac{4\,{a}^{3}bf{x}^{11}}{11}}+{\frac{2\,{a}^{3}be{x}^{10}}{5}}+{\frac{4\,{a}^{3}bd{x}^{9}}{9}}+{\frac{c{a}^{3}b{x}^{8}}{2}}+{\frac{{a}^{4}f{x}^{7}}{7}}+{\frac{{a}^{4}e{x}^{6}}{6}}+{\frac{{a}^{4}d{x}^{5}}{5}}+{\frac{{a}^{4}c{x}^{4}}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x)
[Out]
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Maxima [A] time = 1.37761, size = 271, normalized size = 1.37 \[ \frac{1}{23} \, b^{4} f x^{23} + \frac{1}{22} \, b^{4} e x^{22} + \frac{1}{21} \, b^{4} d x^{21} + \frac{1}{20} \, b^{4} c x^{20} + \frac{4}{19} \, a b^{3} f x^{19} + \frac{2}{9} \, a b^{3} e x^{18} + \frac{4}{17} \, a b^{3} d x^{17} + \frac{1}{4} \, a b^{3} c x^{16} + \frac{2}{5} \, a^{2} b^{2} f x^{15} + \frac{3}{7} \, a^{2} b^{2} e x^{14} + \frac{6}{13} \, a^{2} b^{2} d x^{13} + \frac{1}{2} \, a^{2} b^{2} c x^{12} + \frac{4}{11} \, a^{3} b f x^{11} + \frac{2}{5} \, a^{3} b e x^{10} + \frac{4}{9} \, a^{3} b d x^{9} + \frac{1}{2} \, a^{3} b c x^{8} + \frac{1}{7} \, a^{4} f x^{7} + \frac{1}{6} \, a^{4} e x^{6} + \frac{1}{5} \, a^{4} d x^{5} + \frac{1}{4} \, a^{4} c x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^4*(f*x^3 + e*x^2 + d*x + c)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.187893, size = 1, normalized size = 0.01 \[ \frac{1}{23} x^{23} f b^{4} + \frac{1}{22} x^{22} e b^{4} + \frac{1}{21} x^{21} d b^{4} + \frac{1}{20} x^{20} c b^{4} + \frac{4}{19} x^{19} f b^{3} a + \frac{2}{9} x^{18} e b^{3} a + \frac{4}{17} x^{17} d b^{3} a + \frac{1}{4} x^{16} c b^{3} a + \frac{2}{5} x^{15} f b^{2} a^{2} + \frac{3}{7} x^{14} e b^{2} a^{2} + \frac{6}{13} x^{13} d b^{2} a^{2} + \frac{1}{2} x^{12} c b^{2} a^{2} + \frac{4}{11} x^{11} f b a^{3} + \frac{2}{5} x^{10} e b a^{3} + \frac{4}{9} x^{9} d b a^{3} + \frac{1}{2} x^{8} c b a^{3} + \frac{1}{7} x^{7} f a^{4} + \frac{1}{6} x^{6} e a^{4} + \frac{1}{5} x^{5} d a^{4} + \frac{1}{4} x^{4} c a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^4*(f*x^3 + e*x^2 + d*x + c)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.114462, size = 245, normalized size = 1.24 \[ \frac{a^{4} c x^{4}}{4} + \frac{a^{4} d x^{5}}{5} + \frac{a^{4} e x^{6}}{6} + \frac{a^{4} f x^{7}}{7} + \frac{a^{3} b c x^{8}}{2} + \frac{4 a^{3} b d x^{9}}{9} + \frac{2 a^{3} b e x^{10}}{5} + \frac{4 a^{3} b f x^{11}}{11} + \frac{a^{2} b^{2} c x^{12}}{2} + \frac{6 a^{2} b^{2} d x^{13}}{13} + \frac{3 a^{2} b^{2} e x^{14}}{7} + \frac{2 a^{2} b^{2} f x^{15}}{5} + \frac{a b^{3} c x^{16}}{4} + \frac{4 a b^{3} d x^{17}}{17} + \frac{2 a b^{3} e x^{18}}{9} + \frac{4 a b^{3} f x^{19}}{19} + \frac{b^{4} c x^{20}}{20} + \frac{b^{4} d x^{21}}{21} + \frac{b^{4} e x^{22}}{22} + \frac{b^{4} f x^{23}}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.22603, size = 278, normalized size = 1.4 \[ \frac{1}{23} \, b^{4} f x^{23} + \frac{1}{22} \, b^{4} x^{22} e + \frac{1}{21} \, b^{4} d x^{21} + \frac{1}{20} \, b^{4} c x^{20} + \frac{4}{19} \, a b^{3} f x^{19} + \frac{2}{9} \, a b^{3} x^{18} e + \frac{4}{17} \, a b^{3} d x^{17} + \frac{1}{4} \, a b^{3} c x^{16} + \frac{2}{5} \, a^{2} b^{2} f x^{15} + \frac{3}{7} \, a^{2} b^{2} x^{14} e + \frac{6}{13} \, a^{2} b^{2} d x^{13} + \frac{1}{2} \, a^{2} b^{2} c x^{12} + \frac{4}{11} \, a^{3} b f x^{11} + \frac{2}{5} \, a^{3} b x^{10} e + \frac{4}{9} \, a^{3} b d x^{9} + \frac{1}{2} \, a^{3} b c x^{8} + \frac{1}{7} \, a^{4} f x^{7} + \frac{1}{6} \, a^{4} x^{6} e + \frac{1}{5} \, a^{4} d x^{5} + \frac{1}{4} \, a^{4} c x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^4*(f*x^3 + e*x^2 + d*x + c)*x^3,x, algorithm="giac")
[Out]