Optimal. Leaf size=47 \[ -\frac{a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]
[Out]
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Rubi [A] time = 0.0633064, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2 + c*x^4)^2/x^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{3 x^{3}} - \frac{2 a b}{x} + \frac{2 b c x^{3}}{3} + \frac{c^{2} x^{5}}{5} + \frac{\left (2 a c + b^{2}\right ) \int b^{2}\, dx}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2+a)**2/x**4,x)
[Out]
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Mathematica [A] time = 0.0329877, size = 47, normalized size = 1. \[ -\frac{a^2}{3 x^3}+x \left (2 a c+b^2\right )-\frac{2 a b}{x}+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2 + c*x^4)^2/x^4,x]
[Out]
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Maple [A] time = 0.008, size = 42, normalized size = 0.9 \[{\frac{{c}^{2}{x}^{5}}{5}}+{\frac{2\,bc{x}^{3}}{3}}+2\,xac+{b}^{2}x-{\frac{{a}^{2}}{3\,{x}^{3}}}-2\,{\frac{ab}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2+a)^2/x^4,x)
[Out]
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Maxima [A] time = 0.682364, size = 57, normalized size = 1.21 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} +{\left (b^{2} + 2 \, a c\right )} x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24771, size = 62, normalized size = 1.32 \[ \frac{3 \, c^{2} x^{8} + 10 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 30 \, a b x^{2} - 5 \, a^{2}}{15 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2/x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.17521, size = 44, normalized size = 0.94 \[ \frac{2 b c x^{3}}{3} + \frac{c^{2} x^{5}}{5} + x \left (2 a c + b^{2}\right ) - \frac{a^{2} + 6 a b x^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2+a)**2/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.262874, size = 57, normalized size = 1.21 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} + b^{2} x + 2 \, a c x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2/x^4,x, algorithm="giac")
[Out]