Optimal. Leaf size=94 \[ -\frac{1}{10} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{59}{80} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{1947}{320} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{21417 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.0956151, 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 \[ -\frac{1}{10} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{59}{80} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{1947}{320} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{21417 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x],x]
[Out]
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Rubi in Sympy [A] time = 8.833, size = 83, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{10} - \frac{59 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{80} - \frac{1947 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{320} + \frac{21417 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{3200} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)*(3+5*x)**(3/2)/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.063317, size = 60, normalized size = 0.64 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (800 x^2+2140 x+2943\right )-21417 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3200} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x],x]
[Out]
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Maple [A] time = 0.013, size = 87, normalized size = 0.9 \[{\frac{1}{6400}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -16000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+21417\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -42800\,x\sqrt{-10\,{x}^{2}-x+3}-58860\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)*(3+5*x)^(3/2)/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50554, size = 78, normalized size = 0.83 \[ -\frac{5}{2} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{107}{16} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{21417}{6400} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{2943}{320} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216451, size = 84, normalized size = 0.89 \[ -\frac{1}{6400} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (800 \, x^{2} + 2140 \, x + 2943\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 21417 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 34.0048, size = 224, normalized size = 2.38 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} + \frac{6 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{3 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{1936} + \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)*(3+5*x)**(3/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227513, size = 73, normalized size = 0.78 \[ -\frac{1}{3200} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x + 83\right )}{\left (5 \, x + 3\right )} + 1947\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 21417 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="giac")
[Out]