Optimal. Leaf size=56 \[ \frac{1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac{23}{3} x \sqrt{3 x^2+2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0486342, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac{23}{3} x \sqrt{3 x^2+2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(5 - x)*(3 + 2*x)*Sqrt[2 + 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 5.38462, size = 49, normalized size = 0.88 \[ \frac{23 x \sqrt{3 x^{2} + 2}}{3} + \frac{\left (- 6 x + 28\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{46 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3+2*x)*(3*x**2+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0407873, size = 50, normalized size = 0.89 \[ \frac{1}{18} \left (92 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (9 x^3-42 x^2-132 x-28\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(5 - x)*(3 + 2*x)*Sqrt[2 + 3*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 49, normalized size = 0.9 \[{\frac{23\,x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{46\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{6} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(2*x+3)*(3*x^2+2)^(1/2),x)
[Out]
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Maxima [A] time = 0.776287, size = 65, normalized size = 1.16 \[ -\frac{1}{6} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{7}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{23}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{46}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 2)*(2*x + 3)*(x - 5),x, algorithm="maxima")
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Fricas [A] time = 0.290133, size = 84, normalized size = 1.5 \[ -\frac{1}{54} \, \sqrt{3}{\left (\sqrt{3}{\left (9 \, x^{3} - 42 \, x^{2} - 132 \, x - 28\right )} \sqrt{3 \, x^{2} + 2} - 138 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 2)*(2*x + 3)*(x - 5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.253, size = 94, normalized size = 1.68 \[ - \frac{3 x^{5}}{2 \sqrt{3 x^{2} + 2}} - \frac{3 x^{3}}{2 \sqrt{3 x^{2} + 2}} + \frac{15 x \sqrt{3 x^{2} + 2}}{2} - \frac{x}{3 \sqrt{3 x^{2} + 2}} + \frac{7 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9} + \frac{46 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3+2*x)*(3*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.286304, size = 65, normalized size = 1.16 \[ -\frac{1}{18} \,{\left (3 \,{\left ({\left (3 \, x - 14\right )} x - 44\right )} x - 28\right )} \sqrt{3 \, x^{2} + 2} - \frac{46}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 2)*(2*x + 3)*(x - 5),x, algorithm="giac")
[Out]