Optimal. Leaf size=57 \[ \frac{1}{3} \sqrt{3 x^2+5 x+2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{6 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0447605, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{3} \sqrt{3 x^2+5 x+2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{6 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[2 + 5*x + 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 4.50692, size = 49, normalized size = 0.86 \[ \frac{\sqrt{3 x^{2} + 5 x + 2}}{3} - \frac{5 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(3*x**2+5*x+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0379986, size = 50, normalized size = 0.88 \[ \frac{1}{18} \left (6 \sqrt{3 x^2+5 x+2}-5 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[2 + 5*x + 3*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 45, normalized size = 0.8 \[{\frac{1}{3}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{5\,\sqrt{3}}{18}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(3*x^2+5*x+2)^(1/2),x)
[Out]
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Maxima [A] time = 0.752923, size = 58, normalized size = 1.02 \[ -\frac{5}{18} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(3*x^2 + 5*x + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223354, size = 81, normalized size = 1.42 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 5 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(3*x^2 + 5*x + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(3*x**2+5*x+2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213239, size = 66, normalized size = 1.16 \[ \frac{5}{18} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(3*x^2 + 5*x + 2),x, algorithm="giac")
[Out]