Optimal. Leaf size=116 \[ -\frac{3}{40} \sqrt{1-2 x} (5 x+3)^{7/2}-\frac{247}{480} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{2717}{768} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{29887 \sqrt{1-2 x} \sqrt{5 x+3}}{1024}+\frac{328757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1024 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.119473, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3}{40} \sqrt{1-2 x} (5 x+3)^{7/2}-\frac{247}{480} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{2717}{768} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{29887 \sqrt{1-2 x} \sqrt{5 x+3}}{1024}+\frac{328757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1024 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
[Out]
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Rubi in Sympy [A] time = 10.083, size = 105, normalized size = 0.91 \[ - \frac{3 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{40} - \frac{247 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{480} - \frac{2717 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{768} - \frac{29887 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1024} + \frac{328757 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{10240} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0796354, size = 65, normalized size = 0.56 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (28800 x^3+91360 x^2+132868 x+142713\right )-986271 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{30720} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
[Out]
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Maple [A] time = 0.013, size = 104, normalized size = 0.9 \[{\frac{1}{61440}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1827200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+986271\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -2657360\,x\sqrt{-10\,{x}^{2}-x+3}-2854260\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.51022, size = 101, normalized size = 0.87 \[ -\frac{75}{8} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{2855}{96} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{33217}{768} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{328757}{20480} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{47571}{1024} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218309, size = 90, normalized size = 0.78 \[ -\frac{1}{61440} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (28800 \, x^{3} + 91360 \, x^{2} + 132868 \, x + 142713\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 986271 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 81.2365, size = 298, normalized size = 2.57 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{3 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{1936} + \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} + \frac{6 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (\frac{7 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{3872} + \frac{2 \sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232411, size = 85, normalized size = 0.73 \[ -\frac{1}{30720} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (36 \, x + 71\right )}{\left (5 \, x + 3\right )} + 2717\right )}{\left (5 \, x + 3\right )} + 89661\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 986271 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)/sqrt(-2*x + 1),x, algorithm="giac")
[Out]