Optimal. Leaf size=72 \[ -\frac{1}{4} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{33}{16} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16 \sqrt{10}} \]
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Rubi [A] time = 0.0643038, 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 \[ -\frac{1}{4} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{33}{16} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(3/2)/Sqrt[1 - 2*x],x]
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Rubi in Sympy [A] time = 6.48045, size = 63, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{4} - \frac{33 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{16} + \frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{160} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(3/2)/(1-2*x)**(1/2),x)
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Mathematica [A] time = 0.0481386, size = 55, normalized size = 0.76 \[ \frac{1}{160} \left (-50 \sqrt{1-2 x} \sqrt{5 x+3} (4 x+9)-363 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(3/2)/Sqrt[1 - 2*x],x]
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Maple [A] time = 0.005, size = 72, normalized size = 1. \[ -{\frac{1}{4} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{33}{16}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{363\,\sqrt{10}}{320}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(3/2)/(1-2*x)^(1/2),x)
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Maxima [A] time = 1.50983, size = 55, normalized size = 0.76 \[ -\frac{5}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{363}{320} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{45}{16} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/sqrt(-2*x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.222764, size = 77, normalized size = 1.07 \[ -\frac{1}{320} \, \sqrt{10}{\left (10 \, \sqrt{10} \sqrt{5 \, x + 3}{\left (4 \, x + 9\right )} \sqrt{-2 \, x + 1} - 363 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/sqrt(-2*x + 1),x, algorithm="fricas")
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Sympy [A] time = 6.1716, size = 187, normalized size = 2.6 \[ \begin{cases} - \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{2 \sqrt{10 x - 5}} - \frac{55 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{8 \sqrt{10 x - 5}} + \frac{363 i \sqrt{x + \frac{3}{5}}}{16 \sqrt{10 x - 5}} - \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{160} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{160} + \frac{25 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{2 \sqrt{- 10 x + 5}} + \frac{55 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{8 \sqrt{- 10 x + 5}} - \frac{363 \sqrt{x + \frac{3}{5}}}{16 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)**(3/2)/(1-2*x)**(1/2),x)
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GIAC/XCAS [A] time = 0.228622, size = 61, normalized size = 0.85 \[ -\frac{1}{160} \, \sqrt{5}{\left (10 \, \sqrt{5 \, x + 3}{\left (4 \, x + 9\right )} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/sqrt(-2*x + 1),x, algorithm="giac")
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