Optimal. Leaf size=130 \[ \frac{1}{9} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{5}{24} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{925}{864} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{6553 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2592}+\frac{2}{81} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.30971, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{1}{9} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{5}{24} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{925}{864} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{6553 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2592}+\frac{2}{81} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x),x]
[Out]
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Rubi in Sympy [A] time = 30.3891, size = 117, normalized size = 0.9 \[ \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{9} - \frac{5 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{24} - \frac{925 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{864} + \frac{6553 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{5184} + \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x),x)
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Mathematica [A] time = 0.166796, size = 105, normalized size = 0.81 \[ \frac{12 \sqrt{1-2 x} \sqrt{5 x+3} \left (2400 x^2+1980 x-601\right )+128 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+6553 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{10368} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x),x]
[Out]
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Maple [A] time = 0.013, size = 115, normalized size = 0.9 \[ -{\frac{1}{10368}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -28800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+128\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -6553\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -23760\,x\sqrt{-10\,{x}^{2}-x+3}+7212\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x),x)
[Out]
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Maxima [A] time = 1.51101, size = 112, normalized size = 0.86 \[ -\frac{5}{18} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{145}{72} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{6553}{10368} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{81} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{119}{864} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2),x, algorithm="maxima")
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Fricas [A] time = 0.227346, size = 136, normalized size = 1.05 \[ \frac{1}{10368} \, \sqrt{2}{\left (6 \, \sqrt{2}{\left (2400 \, x^{2} + 1980 \, x - 601\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 64 \, \sqrt{7} \sqrt{2} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6553 \, \sqrt{5} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x),x)
[Out]
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GIAC/XCAS [A] time = 0.290215, size = 251, normalized size = 1.93 \[ -\frac{1}{810} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{4320} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 15 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 925 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{6553}{10368} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2),x, algorithm="giac")
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