Optimal. Leaf size=100 \[ \frac{1}{24} \sqrt{3 x^2+5 x+2} (73-6 x)-\frac{311 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]
[Out]
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Rubi [A] time = 0.178318, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1}{24} \sqrt{3 x^2+5 x+2} (73-6 x)-\frac{311 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x),x]
[Out]
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Rubi in Sympy [A] time = 24.7595, size = 92, normalized size = 0.92 \[ \frac{\left (- 6 x + 73\right ) \sqrt{3 x^{2} + 5 x + 2}}{24} - \frac{311 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{144} - \frac{13 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x),x)
[Out]
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Mathematica [A] time = 0.0791446, size = 114, normalized size = 1.14 \[ \frac{1}{144} \left (-36 \sqrt{3 x^2+5 x+2} x+438 \sqrt{3 x^2+5 x+2}-234 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-311 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )+234 \sqrt{5} \log (2 x+3)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x),x]
[Out]
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Maple [A] time = 0.026, size = 127, normalized size = 1.3 \[ -{\frac{5+6\,x}{24}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{\sqrt{3}}{144}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{8}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13\,\sqrt{3}}{6}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{13\,\sqrt{5}}{8}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x),x)
[Out]
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Maxima [A] time = 0.776889, size = 134, normalized size = 1.34 \[ -\frac{1}{4} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{311}{144} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{13}{8} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{73}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286586, size = 161, normalized size = 1.61 \[ -\frac{1}{288} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 73\right )} - 78 \, \sqrt{5} \sqrt{3} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 311 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x),x)
[Out]
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GIAC/XCAS [A] time = 0.30223, size = 170, normalized size = 1.7 \[ -\frac{1}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 73\right )} + \frac{13}{8} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{311}{144} \, \sqrt{3}{\rm ln}\left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3),x, algorithm="giac")
[Out]