Optimal. Leaf size=94 \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{80} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{407}{800} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{4477 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.0947242, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{80} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{407}{800} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{4477 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x],x]
[Out]
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Rubi in Sympy [A] time = 8.17229, size = 83, normalized size = 0.88 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{10} + \frac{37 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{200} - \frac{407 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{800} + \frac{4477 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0800933, size = 60, normalized size = 0.64 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (800 x^2+820 x-203\right )-4477 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{8000} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x],x]
[Out]
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Maple [A] time = 0.011, size = 87, normalized size = 0.9 \[{\frac{1}{16000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 16000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4477\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +16400\,x\sqrt{-10\,{x}^{2}-x+3}-4060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.47948, size = 74, normalized size = 0.79 \[ -\frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{37}{40} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4477}{16000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{37}{800} \, \sqrt{-10 \, x^{2} - x + 3} \]
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")
[Out]
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Fricas [A] time = 0.217308, size = 84, normalized size = 0.89 \[ \frac{1}{16000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (800 \, x^{2} + 820 \, x - 203\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4477 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
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 = 48.6547, size = 168, normalized size = 1.79 \[ - \frac{7 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{121} + \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\right )}{200} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (- \frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.260515, size = 135, normalized size = 1.44 \[ \frac{1}{8000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
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]