Optimal. Leaf size=94 \[ -\frac{1}{10} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{99}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1089 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.0959091, 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 \[ -\frac{1}{10} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{99}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1089 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x))/Sqrt[3 + 5*x],x]
[Out]
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Rubi in Sympy [A] time = 8.20896, size = 83, normalized size = 0.88 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{10} + \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{40} + \frac{99 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{400} + \frac{1089 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0938027, size = 60, normalized size = 0.64 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-160 x^2+100 x+89\right )-1089 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4000} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(2 + 3*x))/Sqrt[3 + 5*x],x]
[Out]
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Maple [A] time = 0.013, size = 87, normalized size = 0.9 \[{\frac{1}{8000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -3200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1089\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2000\,x\sqrt{-10\,{x}^{2}-x+3}+1780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.49544, size = 78, normalized size = 0.83 \[ -\frac{2}{5} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{1}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1089}{8000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{89}{400} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)*(-2*x + 1)^(3/2)/sqrt(5*x + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217047, size = 84, normalized size = 0.89 \[ -\frac{1}{8000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (160 \, x^{2} - 100 \, x - 89\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1089 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)*(-2*x + 1)^(3/2)/sqrt(5*x + 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 33.6418, size = 223, normalized size = 2.37 \[ - \frac{7 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(3/2)*(2+3*x)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232541, size = 189, normalized size = 2.01 \[ -\frac{1}{20000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{2000} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)*(-2*x + 1)^(3/2)/sqrt(5*x + 3),x, algorithm="giac")
[Out]