Optimal. Leaf size=62 \[ -\frac{2 (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0700811, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{2 (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(3 + 2*x))/(2 + 5*x + 3*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.95299, size = 56, normalized size = 0.9 \[ - \frac{2 \left (139 x + 121\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3+2*x)/(3*x**2+5*x+2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0928223, size = 53, normalized size = 0.85 \[ -\frac{2}{9} \left (\frac{417 x+363}{\sqrt{3 x^2+5 x+2}}+\sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(3 + 2*x))/(2 + 5*x + 3*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 79, normalized size = 1.3 \[ -{\frac{700+840\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{26}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{2\,x}{3}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{2\,\sqrt{3}}{9}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3+2*x)/(3*x^2+5*x+2)^(3/2),x)
[Out]
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Maxima [A] time = 0.7974, size = 78, normalized size = 1.26 \[ -\frac{2}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{278 \, x}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{242}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x + 3)*(x - 5)/(3*x^2 + 5*x + 2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283292, size = 117, normalized size = 1.89 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (139 \, x + 121\right )} -{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x + 3)*(x - 5)/(3*x^2 + 5*x + 2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{7 x}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{2 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{15}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3+2*x)/(3*x**2+5*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.278178, size = 73, normalized size = 1.18 \[ \frac{2}{9} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{2 \,{\left (139 \, x + 121\right )}}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x + 3)*(x - 5)/(3*x^2 + 5*x + 2)^(3/2),x, algorithm="giac")
[Out]