Optimal. Leaf size=154 \[ a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11} \]
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Rubi [A] time = 0.283172, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ B a^{2} \int x\, dx + \frac{B a b x^{4}}{2} + \frac{B b c x^{8}}{4} + \frac{B c^{2} x^{10}}{10} + \frac{B x^{6} \left (2 a c + b^{2}\right )}{6} + \frac{C c^{2} x^{11}}{11} + a^{2} \int A\, dx + \frac{a x^{3} \left (2 A b + C a\right )}{3} + \frac{c x^{9} \left (A c + 2 C b\right )}{9} + x^{7} \left (\frac{2 A b c}{7} + \frac{2 C a c}{7} + \frac{C b^{2}}{7}\right ) + x^{5} \left (\frac{2 A a c}{5} + \frac{A b^{2}}{5} + \frac{2 C a b}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*x**2+B*x+A)*(c*x**4+b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0567707, size = 154, normalized size = 1. \[ a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left (2 a c C+2 A b c+b^2 C\right )+\frac{1}{5} x^5 \left (2 a A c+2 a b C+A b^2\right )+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.001, size = 139, normalized size = 0.9 \[{\frac{{c}^{2}C{x}^{11}}{11}}+{\frac{B{c}^{2}{x}^{10}}{10}}+{\frac{ \left ( A{c}^{2}+2\,Cbc \right ){x}^{9}}{9}}+{\frac{bBc{x}^{8}}{4}}+{\frac{ \left ( 2\,Abc+ \left ( 2\,ac+{b}^{2} \right ) C \right ){x}^{7}}{7}}+{\frac{B \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A \left ( 2\,ac+{b}^{2} \right ) +2\,abC \right ){x}^{5}}{5}}+{\frac{abB{x}^{4}}{2}}+{\frac{ \left ( 2\,abA+{a}^{2}C \right ){x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{2}}{2}}+{a}^{2}Ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+B*x+A)*(c*x^4+b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.697516, size = 189, normalized size = 1.23 \[ \frac{1}{11} \, C c^{2} x^{11} + \frac{1}{10} \, B c^{2} x^{10} + \frac{1}{4} \, B b c x^{8} + \frac{1}{9} \,{\left (2 \, C b c + A c^{2}\right )} x^{9} + \frac{1}{7} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{7} + \frac{1}{2} \, B a b x^{4} + \frac{1}{6} \,{\left (B b^{2} + 2 \, B a c\right )} x^{6} + \frac{1}{5} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x + \frac{1}{3} \,{\left (C a^{2} + 2 \, A a b\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(C*x^2 + B*x + A),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231253, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} c^{2} C + \frac{1}{10} x^{10} c^{2} B + \frac{2}{9} x^{9} c b C + \frac{1}{9} x^{9} c^{2} A + \frac{1}{4} x^{8} c b B + \frac{1}{7} x^{7} b^{2} C + \frac{2}{7} x^{7} c a C + \frac{2}{7} x^{7} c b A + \frac{1}{6} x^{6} b^{2} B + \frac{1}{3} x^{6} c a B + \frac{2}{5} x^{5} b a C + \frac{1}{5} x^{5} b^{2} A + \frac{2}{5} x^{5} c a A + \frac{1}{2} x^{4} b a B + \frac{1}{3} x^{3} a^{2} C + \frac{2}{3} x^{3} b a A + \frac{1}{2} x^{2} a^{2} B + x a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(C*x^2 + B*x + A),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.168474, size = 165, normalized size = 1.07 \[ A a^{2} x + \frac{B a^{2} x^{2}}{2} + \frac{B a b x^{4}}{2} + \frac{B b c x^{8}}{4} + \frac{B c^{2} x^{10}}{10} + \frac{C c^{2} x^{11}}{11} + x^{9} \left (\frac{A c^{2}}{9} + \frac{2 C b c}{9}\right ) + x^{7} \left (\frac{2 A b c}{7} + \frac{2 C a c}{7} + \frac{C b^{2}}{7}\right ) + x^{6} \left (\frac{B a c}{3} + \frac{B b^{2}}{6}\right ) + x^{5} \left (\frac{2 A a c}{5} + \frac{A b^{2}}{5} + \frac{2 C a b}{5}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{C a^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x**2+B*x+A)*(c*x**4+b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.27819, size = 204, normalized size = 1.32 \[ \frac{1}{11} \, C c^{2} x^{11} + \frac{1}{10} \, B c^{2} x^{10} + \frac{2}{9} \, C b c x^{9} + \frac{1}{9} \, A c^{2} x^{9} + \frac{1}{4} \, B b c x^{8} + \frac{1}{7} \, C b^{2} x^{7} + \frac{2}{7} \, C a c x^{7} + \frac{2}{7} \, A b c x^{7} + \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, B a c x^{6} + \frac{2}{5} \, C a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{2}{5} \, A a c x^{5} + \frac{1}{2} \, B a b x^{4} + \frac{1}{3} \, C a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(C*x^2 + B*x + A),x, algorithm="giac")
[Out]