Optimal. Leaf size=156 \[ -\frac{a^3 A}{x}+a^2 x (a B+3 A b)+\frac{1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13} \]
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Rubi [A] time = 0.27807, antiderivative size = 156, 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 \[ -\frac{a^3 A}{x}+a^2 x (a B+3 A b)+\frac{1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{3}}{x} + \frac{B c^{3} x^{13}}{13} + a x^{3} \left (A a c + A b^{2} + B a b\right ) + \frac{c^{2} x^{11} \left (A c + 3 B b\right )}{11} + \frac{c x^{9} \left (A b c + B a c + B b^{2}\right )}{3} + x^{7} \left (\frac{3 A a c^{2}}{7} + \frac{3 A b^{2} c}{7} + \frac{6 B a b c}{7} + \frac{B b^{3}}{7}\right ) + x^{5} \left (\frac{6 A a b c}{5} + \frac{A b^{3}}{5} + \frac{3 B a^{2} c}{5} + \frac{3 B a b^{2}}{5}\right ) + \frac{a^{2} \left (3 A b + B a\right ) \int B\, dx}{B} \]
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**2,x)
[Out]
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Mathematica [A] time = 0.16053, size = 156, normalized size = 1. \[ -\frac{a^3 A}{x}+a^2 x (a B+3 A b)+\frac{1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^2,x]
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Maple [A] time = 0.006, size = 186, normalized size = 1.2 \[{\frac{B{c}^{3}{x}^{13}}{13}}+{\frac{A{x}^{11}{c}^{3}}{11}}+{\frac{3\,B{x}^{11}b{c}^{2}}{11}}+{\frac{A{x}^{9}b{c}^{2}}{3}}+{\frac{B{x}^{9}a{c}^{2}}{3}}+{\frac{B{x}^{9}{b}^{2}c}{3}}+{\frac{3\,A{x}^{7}a{c}^{2}}{7}}+{\frac{3\,A{x}^{7}{b}^{2}c}{7}}+{\frac{6\,B{x}^{7}abc}{7}}+{\frac{B{x}^{7}{b}^{3}}{7}}+{\frac{6\,A{x}^{5}abc}{5}}+{\frac{A{x}^{5}{b}^{3}}{5}}+{\frac{3\,B{x}^{5}{a}^{2}c}{5}}+{\frac{3\,B{x}^{5}a{b}^{2}}{5}}+A{x}^{3}{a}^{2}c+A{x}^{3}a{b}^{2}+B{x}^{3}{a}^{2}b+3\,Ax{a}^{2}b+Bx{a}^{3}-{\frac{A{a}^{3}}{x}} \]
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^2,x)
[Out]
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Maxima [A] time = 0.699918, size = 219, normalized size = 1.4 \[ \frac{1}{13} \, B c^{3} x^{13} + \frac{1}{11} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{11} + \frac{1}{3} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{9} + \frac{1}{7} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + \frac{1}{5} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{5} +{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} - \frac{A a^{3}}{x} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \]
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^2,x, algorithm="maxima")
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Fricas [A] time = 0.241027, size = 227, normalized size = 1.46 \[ \frac{1155 \, B c^{3} x^{14} + 1365 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + 5005 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{10} + 2145 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 3003 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 15015 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} - 15015 \, A a^{3} + 15015 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}}{15015 \, x} \]
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^2,x, algorithm="fricas")
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Sympy [A] time = 1.73799, size = 185, normalized size = 1.19 \[ - \frac{A a^{3}}{x} + \frac{B c^{3} x^{13}}{13} + x^{11} \left (\frac{A c^{3}}{11} + \frac{3 B b c^{2}}{11}\right ) + x^{9} \left (\frac{A b c^{2}}{3} + \frac{B a c^{2}}{3} + \frac{B b^{2} c}{3}\right ) + x^{7} \left (\frac{3 A a c^{2}}{7} + \frac{3 A b^{2} c}{7} + \frac{6 B a b c}{7} + \frac{B b^{3}}{7}\right ) + x^{5} \left (\frac{6 A a b c}{5} + \frac{A b^{3}}{5} + \frac{3 B a^{2} c}{5} + \frac{3 B a b^{2}}{5}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x \left (3 A a^{2} b + B a^{3}\right ) \]
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**2,x)
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GIAC/XCAS [A] time = 0.262282, size = 250, normalized size = 1.6 \[ \frac{1}{13} \, B c^{3} x^{13} + \frac{3}{11} \, B b c^{2} x^{11} + \frac{1}{11} \, A c^{3} x^{11} + \frac{1}{3} \, B b^{2} c x^{9} + \frac{1}{3} \, B a c^{2} x^{9} + \frac{1}{3} \, A b c^{2} x^{9} + \frac{1}{7} \, B b^{3} x^{7} + \frac{6}{7} \, B a b c x^{7} + \frac{3}{7} \, A b^{2} c x^{7} + \frac{3}{7} \, A a c^{2} x^{7} + \frac{3}{5} \, B a b^{2} x^{5} + \frac{1}{5} \, A b^{3} x^{5} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{6}{5} \, A a b c x^{5} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + B a^{3} x + 3 \, A a^{2} b x - \frac{A a^{3}}{x} \]
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^2,x, algorithm="giac")
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