Optimal. Leaf size=43 \[ -\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}} \]
[Out]
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Rubi [A] time = 0.0488336, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 9.56137, size = 37, normalized size = 0.86 \[ - \frac{33 x + 55}{46 \left (2 x^{2} - x + 3\right )} + \frac{82 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{529} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+3*x+2)/(2*x**2-x+3)**2,x)
[Out]
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Mathematica [A] time = 0.0302141, size = 43, normalized size = 1. \[ \frac{82 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{23 \sqrt{23}}-\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^2,x]
[Out]
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Maple [A] time = 0.007, size = 34, normalized size = 0.8 \[{1 \left ( -{\frac{33\,x}{92}}-{\frac{55}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{82\,\sqrt{23}}{529}\arctan \left ({\frac{ \left ( 8\,x-2 \right ) \sqrt{23}}{46}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+3*x+2)/(2*x^2-x+3)^2,x)
[Out]
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Maxima [A] time = 0.765322, size = 49, normalized size = 1.14 \[ \frac{82}{529} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{46 \,{\left (2 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258337, size = 69, normalized size = 1.6 \[ \frac{\sqrt{23}{\left (164 \,{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 11 \, \sqrt{23}{\left (3 \, x + 5\right )}\right )}}{1058 \,{\left (2 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.170158, size = 41, normalized size = 0.95 \[ - \frac{33 x + 55}{92 x^{2} - 46 x + 138} + \frac{82 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{529} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+3*x+2)/(2*x**2-x+3)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.26524, size = 49, normalized size = 1.14 \[ \frac{82}{529} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{46 \,{\left (2 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3)^2,x, algorithm="giac")
[Out]