Optimal. Leaf size=94 \[ -\frac{9}{100} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{2 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}+\frac{317 \sqrt{5 x+3} \sqrt{1-2 x}}{2200}+\frac{317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.114583, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{9}{100} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{2 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}+\frac{317 \sqrt{5 x+3} \sqrt{1-2 x}}{2200}+\frac{317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(2 + 3*x)^2)/(3 + 5*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.71293, size = 85, normalized size = 0.9 \[ - \frac{9 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{100} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{275 \sqrt{5 x + 3}} + \frac{317 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2200} + \frac{317 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**2*(1-2*x)**(1/2)/(3+5*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.138073, size = 60, normalized size = 0.64 \[ \frac{\frac{10 \sqrt{1-2 x} \left (180 x^2+165 x+31\right )}{\sqrt{5 x+3}}-317 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^2)/(3 + 5*x)^(3/2),x]
[Out]
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Maple [A] time = 0.019, size = 99, normalized size = 1.1 \[{\frac{1}{4000} \left ( 1585\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+3600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+951\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3300\,x\sqrt{-10\,{x}^{2}-x+3}+620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.51334, size = 88, normalized size = 0.94 \[ \frac{317}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{50} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{57}{1000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{125 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2*sqrt(-2*x + 1)/(5*x + 3)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217571, size = 100, normalized size = 1.06 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (180 \, x^{2} + 165 \, x + 31\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 317 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4000 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2*sqrt(-2*x + 1)/(5*x + 3)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{\left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**2*(1-2*x)**(1/2)/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270085, size = 150, normalized size = 1.6 \[ \frac{3}{5000} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 17 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{317}{2000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2*sqrt(-2*x + 1)/(5*x + 3)^(3/2),x, algorithm="giac")
[Out]