Optimal. Leaf size=72 \[ -\frac{1}{4} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{11}{40} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]
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Rubi [A] time = 0.0628127, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{4} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{11}{40} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]*Sqrt[3 + 5*x],x]
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Rubi in Sympy [A] time = 6.47012, size = 63, normalized size = 0.88 \[ \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{10} - \frac{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{40} + \frac{121 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{400} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(1/2)*(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0419034, size = 55, normalized size = 0.76 \[ \frac{1}{400} \left (10 \sqrt{1-2 x} \sqrt{5 x+3} (20 x+1)-121 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]*Sqrt[3 + 5*x],x]
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Maple [A] time = 0.008, size = 72, normalized size = 1. \[{\frac{1}{10} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{11}{40}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{121\,\sqrt{10}}{800}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(1/2)*(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50114, size = 55, normalized size = 0.76 \[ \frac{1}{2} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{121}{800} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1}{40} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(-2*x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.214407, size = 77, normalized size = 1.07 \[ \frac{1}{800} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 121 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(-2*x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.33179, size = 184, normalized size = 2.56 \[ \begin{cases} \frac{5 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{10 x - 5}} - \frac{33 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4 \sqrt{10 x - 5}} + \frac{121 i \sqrt{x + \frac{3}{5}}}{40 \sqrt{10 x - 5}} - \frac{121 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{400} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{121 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{400} - \frac{5 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{- 10 x + 5}} + \frac{33 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4 \sqrt{- 10 x + 5}} - \frac{121 \sqrt{x + \frac{3}{5}}}{40 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(1/2)*(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.244136, size = 61, normalized size = 0.85 \[ \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(-2*x + 1),x, algorithm="giac")
[Out]