Optimal. Leaf size=63 \[ \frac{121 (19-7 x)}{184 \left (2 x^2-x+3\right )}+\frac{55}{8} \log \left (2 x^2-x+3\right )+\frac{25 x}{4}+\frac{1859 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{92 \sqrt{23}} \]
[Out]
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Rubi [A] time = 0.110354, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{121 (19-7 x)}{184 \left (2 x^2-x+3\right )}+\frac{55}{8} \log \left (2 x^2-x+3\right )+\frac{25 x}{4}+\frac{1859 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{92 \sqrt{23}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{50 x^{3}}{23} - \frac{\left (- 4 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{2}}{23 \left (2 x^{2} - x + 3\right )} + \frac{55 \log{\left (2 x^{2} - x + 3 \right )}}{8} - \frac{1859 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{2116} - \frac{\int \left (- \frac{279}{2}\right )\, dx}{23} - \frac{145 \int x\, dx}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+3*x+2)**2/(2*x**2-x+3)**2,x)
[Out]
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Mathematica [A] time = 0.0592621, size = 63, normalized size = 1. \[ -\frac{121 (7 x-19)}{184 \left (2 x^2-x+3\right )}+\frac{55}{8} \log \left (2 x^2-x+3\right )+\frac{25 x}{4}-\frac{1859 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{92 \sqrt{23}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^2,x]
[Out]
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Maple [A] time = 0.009, size = 51, normalized size = 0.8 \[{\frac{25\,x}{4}}+{\frac{11}{4} \left ( -{\frac{77\,x}{92}}+{\frac{209}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{55\,\ln \left ( 4\,{x}^{2}-2\,x+6 \right ) }{8}}-{\frac{1859\,\sqrt{23}}{2116}\arctan \left ({\frac{ \left ( 8\,x-2 \right ) \sqrt{23}}{46}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+3*x+2)^2/(2*x^2-x+3)^2,x)
[Out]
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Maxima [A] time = 0.769517, size = 70, normalized size = 1.11 \[ -\frac{1859}{2116} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{25}{4} \, x - \frac{121 \,{\left (7 \, x - 19\right )}}{184 \,{\left (2 \, x^{2} - x + 3\right )}} + \frac{55}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)^2/(2*x^2 - x + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260563, size = 116, normalized size = 1.84 \[ \frac{\sqrt{23}{\left (1265 \, \sqrt{23}{\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) - 3718 \,{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \sqrt{23}{\left (2300 \, x^{3} - 1150 \, x^{2} + 2603 \, x + 2299\right )}\right )}}{4232 \,{\left (2 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)^2/(2*x^2 - x + 3)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.208861, size = 61, normalized size = 0.97 \[ \frac{25 x}{4} - \frac{847 x - 2299}{368 x^{2} - 184 x + 552} + \frac{55 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{8} - \frac{1859 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{2116} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+3*x+2)**2/(2*x**2-x+3)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.265023, size = 70, normalized size = 1.11 \[ -\frac{1859}{2116} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{25}{4} \, x - \frac{121 \,{\left (7 \, x - 19\right )}}{184 \,{\left (2 \, x^{2} - x + 3\right )}} + \frac{55}{8} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)^2/(2*x^2 - x + 3)^2,x, algorithm="giac")
[Out]