Optimal. Leaf size=166 \[ \frac{1}{4} a^3 A x^4+\frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{3}{14} c x^{14} \left (a B c+A b c+b^2 B\right )+\frac{3}{8} a x^8 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{12} x^{12} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{10} x^{10} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{16} c^2 x^{16} (A c+3 b B)+\frac{1}{18} B c^3 x^{18} \]
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Rubi [A] time = 0.775258, antiderivative size = 166, 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 \[ \frac{1}{4} a^3 A x^4+\frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{3}{14} c x^{14} \left (a B c+A b c+b^2 B\right )+\frac{3}{8} a x^8 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{12} x^{12} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{10} x^{10} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{16} c^2 x^{16} (A c+3 b B)+\frac{1}{18} B c^3 x^{18} \]
Antiderivative was successfully verified.
[In] Int[x^3*(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{A a^{3} \int ^{x^{2}} x\, dx}{2} + \frac{B c^{3} x^{18}}{18} + \frac{a^{2} x^{6} \left (3 A b + B a\right )}{6} + \frac{3 a x^{8} \left (A a c + A b^{2} + B a b\right )}{8} + \frac{c^{2} x^{16} \left (A c + 3 B b\right )}{16} + \frac{3 c x^{14} \left (A b c + B a c + B b^{2}\right )}{14} + x^{12} \left (\frac{A a c^{2}}{4} + \frac{A b^{2} c}{4} + \frac{B a b c}{2} + \frac{B b^{3}}{12}\right ) + x^{10} \left (\frac{3 A a b c}{5} + \frac{A b^{3}}{10} + \frac{3 B a^{2} c}{10} + \frac{3 B a b^{2}}{10}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x**2+A)*(c*x**4+b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0839754, size = 166, normalized size = 1. \[ \frac{1}{4} a^3 A x^4+\frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{3}{14} c x^{14} \left (a B c+A b c+b^2 B\right )+\frac{3}{8} a x^8 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{12} x^{12} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{10} x^{10} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{16} c^2 x^{16} (A c+3 b B)+\frac{1}{18} B c^3 x^{18} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]
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Maple [A] time = 0.001, size = 226, normalized size = 1.4 \[{\frac{B{c}^{3}{x}^{18}}{18}}+{\frac{ \left ( A{c}^{3}+3\,B{c}^{2}b \right ){x}^{16}}{16}}+{\frac{ \left ( 3\,A{c}^{2}b+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{14}}{14}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{12}}{12}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{10}}{10}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\,B{a}^{2}b \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{6}}{6}}+{\frac{{a}^{3}A{x}^{4}}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^2+A)*(c*x^4+b*x^2+a)^3,x)
[Out]
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Maxima [A] time = 0.70335, size = 224, normalized size = 1.35 \[ \frac{1}{18} \, B c^{3} x^{18} + \frac{1}{16} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{16} + \frac{3}{14} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{14} + \frac{1}{12} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{12} + \frac{1}{10} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{10} + \frac{3}{8} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{8} + \frac{1}{4} \, A a^{3} x^{4} + \frac{1}{6} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^3*(B*x^2 + A)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24504, size = 1, normalized size = 0.01 \[ \frac{1}{18} x^{18} c^{3} B + \frac{3}{16} x^{16} c^{2} b B + \frac{1}{16} x^{16} c^{3} A + \frac{3}{14} x^{14} c b^{2} B + \frac{3}{14} x^{14} c^{2} a B + \frac{3}{14} x^{14} c^{2} b A + \frac{1}{12} x^{12} b^{3} B + \frac{1}{2} x^{12} c b a B + \frac{1}{4} x^{12} c b^{2} A + \frac{1}{4} x^{12} c^{2} a A + \frac{3}{10} x^{10} b^{2} a B + \frac{3}{10} x^{10} c a^{2} B + \frac{1}{10} x^{10} b^{3} A + \frac{3}{5} x^{10} c b a A + \frac{3}{8} x^{8} b a^{2} B + \frac{3}{8} x^{8} b^{2} a A + \frac{3}{8} x^{8} c a^{2} A + \frac{1}{6} x^{6} a^{3} B + \frac{1}{2} x^{6} b a^{2} A + \frac{1}{4} x^{4} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^3*(B*x^2 + A)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.194115, size = 202, normalized size = 1.22 \[ \frac{A a^{3} x^{4}}{4} + \frac{B c^{3} x^{18}}{18} + x^{16} \left (\frac{A c^{3}}{16} + \frac{3 B b c^{2}}{16}\right ) + x^{14} \left (\frac{3 A b c^{2}}{14} + \frac{3 B a c^{2}}{14} + \frac{3 B b^{2} c}{14}\right ) + x^{12} \left (\frac{A a c^{2}}{4} + \frac{A b^{2} c}{4} + \frac{B a b c}{2} + \frac{B b^{3}}{12}\right ) + x^{10} \left (\frac{3 A a b c}{5} + \frac{A b^{3}}{10} + \frac{3 B a^{2} c}{10} + \frac{3 B a b^{2}}{10}\right ) + x^{8} \left (\frac{3 A a^{2} c}{8} + \frac{3 A a b^{2}}{8} + \frac{3 B a^{2} b}{8}\right ) + x^{6} \left (\frac{A a^{2} b}{2} + \frac{B a^{3}}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**2+A)*(c*x**4+b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.263473, size = 261, normalized size = 1.57 \[ \frac{1}{18} \, B c^{3} x^{18} + \frac{3}{16} \, B b c^{2} x^{16} + \frac{1}{16} \, A c^{3} x^{16} + \frac{3}{14} \, B b^{2} c x^{14} + \frac{3}{14} \, B a c^{2} x^{14} + \frac{3}{14} \, A b c^{2} x^{14} + \frac{1}{12} \, B b^{3} x^{12} + \frac{1}{2} \, B a b c x^{12} + \frac{1}{4} \, A b^{2} c x^{12} + \frac{1}{4} \, A a c^{2} x^{12} + \frac{3}{10} \, B a b^{2} x^{10} + \frac{1}{10} \, A b^{3} x^{10} + \frac{3}{10} \, B a^{2} c x^{10} + \frac{3}{5} \, A a b c x^{10} + \frac{3}{8} \, B a^{2} b x^{8} + \frac{3}{8} \, A a b^{2} x^{8} + \frac{3}{8} \, A a^{2} c x^{8} + \frac{1}{6} \, B a^{3} x^{6} + \frac{1}{2} \, A a^{2} b x^{6} + \frac{1}{4} \, A a^{3} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^3*(B*x^2 + A)*x^3,x, algorithm="giac")
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