Optimal. Leaf size=42 \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}} \]
[Out]
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Rubi [A] time = 0.0672588, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{11 \log{\left (2 x^{2} - x + 3 \right )}}{8} - \frac{33 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{92} + \int \frac{5}{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+3*x+2)/(2*x**2-x+3),x)
[Out]
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Mathematica [A] time = 0.0197414, size = 42, normalized size = 1. \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}-\frac{33 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{4 \sqrt{23}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 34, normalized size = 0.8 \[{\frac{5\,x}{2}}+{\frac{11\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{8}}-{\frac{33\,\sqrt{23}}{92}\arctan \left ({\frac{ \left ( 4\,x-1 \right ) \sqrt{23}}{23}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+3*x+2)/(2*x^2-x+3),x)
[Out]
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Maxima [A] time = 0.769842, size = 45, normalized size = 1.07 \[ -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268168, size = 55, normalized size = 1.31 \[ \frac{1}{184} \, \sqrt{23}{\left (20 \, \sqrt{23} x + 11 \, \sqrt{23} \log \left (2 \, x^{2} - x + 3\right ) - 66 \, \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.135538, size = 46, normalized size = 1.1 \[ \frac{5 x}{2} + \frac{11 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{8} - \frac{33 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{92} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+3*x+2)/(2*x**2-x+3),x)
[Out]
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GIAC/XCAS [A] time = 0.264574, size = 45, normalized size = 1.07 \[ -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3),x, algorithm="giac")
[Out]