Optimal. Leaf size=72 \[ -\frac{3}{20} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{107}{80} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80 \sqrt{10}} \]
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Rubi [A] time = 0.0717162, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3}{20} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{107}{80} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x],x]
[Out]
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Rubi in Sympy [A] time = 6.88674, size = 65, normalized size = 0.9 \[ - \frac{3 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{20} - \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{80} + \frac{1177 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{800} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)*(3+5*x)**(1/2)/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0496415, size = 55, normalized size = 0.76 \[ \frac{1}{800} \left (-10 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+143)-1177 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x],x]
[Out]
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Maple [A] time = 0.013, size = 70, normalized size = 1. \[{\frac{1}{1600}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1177\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1200\,x\sqrt{-10\,{x}^{2}-x+3}-2860\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)*(3+5*x)^(1/2)/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.49729, size = 59, normalized size = 0.82 \[ \frac{1177}{1600} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{143}{80} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.215348, size = 77, normalized size = 1.07 \[ -\frac{1}{1600} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (60 \, x + 143\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1177 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.68106, size = 167, normalized size = 2.32 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} + \frac{6 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)*(3+5*x)**(1/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227266, size = 61, normalized size = 0.85 \[ -\frac{1}{800} \, \sqrt{5}{\left (2 \,{\left (60 \, x + 143\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 1177 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="giac")
[Out]