Optimal. Leaf size=161 \[ a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac{3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15} \]
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Rubi [A] time = 0.302797, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac{3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15} \]
Antiderivative was successfully verified.
[In] Int[(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. \[ \frac{B c^{3} x^{15}}{15} + a^{3} \int A\, dx + \frac{a^{2} x^{3} \left (3 A b + B a\right )}{3} + \frac{3 a x^{5} \left (A a c + A b^{2} + B a b\right )}{5} + \frac{c^{2} x^{13} \left (A c + 3 B b\right )}{13} + \frac{3 c x^{11} \left (A b c + B a c + B b^{2}\right )}{11} + x^{9} \left (\frac{A a c^{2}}{3} + \frac{A b^{2} c}{3} + \frac{2 B a b c}{3} + \frac{B b^{3}}{9}\right ) + x^{7} \left (\frac{6 A a b c}{7} + \frac{A b^{3}}{7} + \frac{3 B a^{2} c}{7} + \frac{3 B a b^{2}}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(c*x**4+b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0896112, size = 161, normalized size = 1. \[ a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac{3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)*(a + b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.001, size = 223, normalized size = 1.4 \[{\frac{B{c}^{3}{x}^{15}}{15}}+{\frac{ \left ( A{c}^{3}+3\,B{c}^{2}b \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,A{c}^{2}b+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{11}}{11}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\,B{a}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{3}}{3}}+{a}^{3}Ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(c*x^4+b*x^2+a)^3,x)
[Out]
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Maxima [A] time = 0.712528, size = 220, normalized size = 1.37 \[ \frac{1}{15} \, B c^{3} x^{15} + \frac{1}{13} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + \frac{3}{11} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{11} + \frac{1}{9} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{9} + \frac{1}{7} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + \frac{3}{5} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + A a^{3} x + \frac{1}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \]
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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239825, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} c^{3} B + \frac{3}{13} x^{13} c^{2} b B + \frac{1}{13} x^{13} c^{3} A + \frac{3}{11} x^{11} c b^{2} B + \frac{3}{11} x^{11} c^{2} a B + \frac{3}{11} x^{11} c^{2} b A + \frac{1}{9} x^{9} b^{3} B + \frac{2}{3} x^{9} c b a B + \frac{1}{3} x^{9} c b^{2} A + \frac{1}{3} x^{9} c^{2} a A + \frac{3}{7} x^{7} b^{2} a B + \frac{3}{7} x^{7} c a^{2} B + \frac{1}{7} x^{7} b^{3} A + \frac{6}{7} x^{7} c b a A + \frac{3}{5} x^{5} b a^{2} B + \frac{3}{5} x^{5} b^{2} a A + \frac{3}{5} x^{5} c a^{2} A + \frac{1}{3} x^{3} a^{3} B + x^{3} b a^{2} A + x a^{3} A \]
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, algorithm="fricas")
[Out]
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Sympy [A] time = 0.186605, size = 199, normalized size = 1.24 \[ A a^{3} x + \frac{B c^{3} x^{15}}{15} + x^{13} \left (\frac{A c^{3}}{13} + \frac{3 B b c^{2}}{13}\right ) + x^{11} \left (\frac{3 A b c^{2}}{11} + \frac{3 B a c^{2}}{11} + \frac{3 B b^{2} c}{11}\right ) + x^{9} \left (\frac{A a c^{2}}{3} + \frac{A b^{2} c}{3} + \frac{2 B a b c}{3} + \frac{B b^{3}}{9}\right ) + x^{7} \left (\frac{6 A a b c}{7} + \frac{A b^{3}}{7} + \frac{3 B a^{2} c}{7} + \frac{3 B a b^{2}}{7}\right ) + x^{5} \left (\frac{3 A a^{2} c}{5} + \frac{3 A a b^{2}}{5} + \frac{3 B a^{2} b}{5}\right ) + x^{3} \left (A a^{2} b + \frac{B a^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)*(c*x**4+b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.262109, size = 255, normalized size = 1.58 \[ \frac{1}{15} \, B c^{3} x^{15} + \frac{3}{13} \, B b c^{2} x^{13} + \frac{1}{13} \, A c^{3} x^{13} + \frac{3}{11} \, B b^{2} c x^{11} + \frac{3}{11} \, B a c^{2} x^{11} + \frac{3}{11} \, A b c^{2} x^{11} + \frac{1}{9} \, B b^{3} x^{9} + \frac{2}{3} \, B a b c x^{9} + \frac{1}{3} \, A b^{2} c x^{9} + \frac{1}{3} \, A a c^{2} x^{9} + \frac{3}{7} \, B a b^{2} x^{7} + \frac{1}{7} \, A b^{3} x^{7} + \frac{3}{7} \, B a^{2} c x^{7} + \frac{6}{7} \, A a b c x^{7} + \frac{3}{5} \, B a^{2} b x^{5} + \frac{3}{5} \, A a b^{2} x^{5} + \frac{3}{5} \, A a^{2} c x^{5} + \frac{1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + A a^{3} x \]
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, algorithm="giac")
[Out]