Optimal. Leaf size=36 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.036912, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[2 + 5*x + 3*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 5.17256, size = 34, normalized size = 0.94 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (5 x + 4\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(3*x**2+5*x+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0346353, size = 33, normalized size = 0.92 \[ \frac{\log (x)-\log \left (2 \sqrt{6 x^2+10 x+4}+5 x+4\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[2 + 5*x + 3*x^2]),x]
[Out]
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Maple [A] time = 0.006, size = 29, normalized size = 0.8 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+5\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(3*x^2+5*x+2)^(1/2),x)
[Out]
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Maxima [A] time = 0.749242, size = 47, normalized size = 1.31 \[ -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(3*x^2 + 5*x + 2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221251, size = 58, normalized size = 1.61 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{4 \, \sqrt{2} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x + 4\right )} - 49 \, x^{2} - 80 \, x - 32}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(3*x^2 + 5*x + 2)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(3*x**2+5*x+2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.233263, size = 82, normalized size = 2.28 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{3} x + \sqrt{2} + \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{3} x - \sqrt{2} + \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(3*x^2 + 5*x + 2)*x),x, algorithm="giac")
[Out]