Optimal. Leaf size=143 \[ -\frac{a^2 A}{5 x^5}-\frac{a^2 B}{4 x^4}+x \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{A \left (2 a c+b^2\right )+2 a b C}{x}-\frac{a (a C+2 A b)}{3 x^3}+B \log (x) \left (2 a c+b^2\right )-\frac{a b B}{x^2}+\frac{1}{3} c x^3 (A c+2 b C)+b B c x^2+\frac{1}{4} B c^2 x^4+\frac{1}{5} c^2 C x^5 \]
[Out]
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Rubi [A] time = 0.340421, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{a^2 A}{5 x^5}-\frac{a^2 B}{4 x^4}+x \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{A \left (2 a c+b^2\right )+2 a b C}{x}-\frac{a (a C+2 A b)}{3 x^3}+B \log (x) \left (2 a c+b^2\right )-\frac{a b B}{x^2}+\frac{1}{3} c x^3 (A c+2 b C)+b B c x^2+\frac{1}{4} B c^2 x^4+\frac{1}{5} c^2 C x^5 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^6,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{5 x^{5}} - \frac{B a^{2}}{4 x^{4}} - \frac{B a b}{x^{2}} + 2 B b c \int x\, dx + \frac{B c^{2} x^{4}}{4} + B \left (2 a c + b^{2}\right ) \log{\left (x \right )} + \frac{C c^{2} x^{5}}{5} - \frac{a \left (2 A b + C a\right )}{3 x^{3}} + \frac{c x^{3} \left (A c + 2 C b\right )}{3} - \frac{2 A a c + A b^{2} + 2 C a b}{x} + \frac{\left (C b^{2} + 2 c \left (A b + C a\right )\right ) \int C\, dx}{C} \]
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**6,x)
[Out]
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Mathematica [A] time = 0.172236, size = 142, normalized size = 0.99 \[ -\frac{a^2 A}{5 x^5}-\frac{a^2 B}{4 x^4}-\frac{2 a A c+2 a b C+A b^2}{x}-\frac{a (a C+2 A b)}{3 x^3}+B \log (x) \left (2 a c+b^2\right )+C x \left (2 a c+b^2\right )-\frac{a b B}{x^2}+\frac{1}{3} c x^3 (A c+2 b C)+2 A b c x+b B c x^2+\frac{1}{4} B c^2 x^4+\frac{1}{5} c^2 C x^5 \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^6,x]
[Out]
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Maple [A] time = 0.012, size = 144, normalized size = 1. \[{\frac{{c}^{2}C{x}^{5}}{5}}+{\frac{B{c}^{2}{x}^{4}}{4}}+{\frac{A{c}^{2}{x}^{3}}{3}}+{\frac{2\,C{x}^{3}bc}{3}}+bBc{x}^{2}+2\,Axbc+2\,Cxac+Cx{b}^{2}-{\frac{2\,abA}{3\,{x}^{3}}}-{\frac{{a}^{2}C}{3\,{x}^{3}}}+2\,B\ln \left ( x \right ) ac+B\ln \left ( x \right ){b}^{2}-2\,{\frac{aAc}{x}}-{\frac{A{b}^{2}}{x}}-2\,{\frac{abC}{x}}-{\frac{abB}{{x}^{2}}}-{\frac{A{a}^{2}}{5\,{x}^{5}}}-{\frac{B{a}^{2}}{4\,{x}^{4}}} \]
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^6,x)
[Out]
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Maxima [A] time = 0.693777, size = 186, normalized size = 1.3 \[ \frac{1}{5} \, C c^{2} x^{5} + \frac{1}{4} \, B c^{2} x^{4} + B b c x^{2} + \frac{1}{3} \,{\left (2 \, C b c + A c^{2}\right )} x^{3} +{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x +{\left (B b^{2} + 2 \, B a c\right )} \log \left (x\right ) - \frac{60 \, B a b x^{3} + 60 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 15 \, B a^{2} x + 12 \, A a^{2} + 20 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{5}} \]
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^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.250123, size = 196, normalized size = 1.37 \[ \frac{12 \, C c^{2} x^{10} + 15 \, B c^{2} x^{9} + 60 \, B b c x^{7} + 20 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 60 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} \log \left (x\right ) - 60 \, B a b x^{3} - 60 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 15 \, B a^{2} x - 12 \, A a^{2} - 20 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{5}} \]
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^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 31.2289, size = 151, normalized size = 1.06 \[ B b c x^{2} + \frac{B c^{2} x^{4}}{4} + B \left (2 a c + b^{2}\right ) \log{\left (x \right )} + \frac{C c^{2} x^{5}}{5} + x^{3} \left (\frac{A c^{2}}{3} + \frac{2 C b c}{3}\right ) + x \left (2 A b c + 2 C a c + C b^{2}\right ) - \frac{12 A a^{2} + 15 B a^{2} x + 60 B a b x^{3} + x^{4} \left (120 A a c + 60 A b^{2} + 120 C a b\right ) + x^{2} \left (40 A a b + 20 C a^{2}\right )}{60 x^{5}} \]
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**6,x)
[Out]
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GIAC/XCAS [A] time = 0.279553, size = 189, normalized size = 1.32 \[ \frac{1}{5} \, C c^{2} x^{5} + \frac{1}{4} \, B c^{2} x^{4} + \frac{2}{3} \, C b c x^{3} + \frac{1}{3} \, A c^{2} x^{3} + B b c x^{2} + C b^{2} x + 2 \, C a c x + 2 \, A b c x +{\left (B b^{2} + 2 \, B a c\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{60 \, B a b x^{3} + 60 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 15 \, B a^{2} x + 12 \, A a^{2} + 20 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{5}} \]
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^6,x, algorithm="giac")
[Out]