Optimal. Leaf size=116 \[ -\frac{1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{5 x+3}+\frac{1331}{512} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512 \sqrt{10}} \]
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Rubi [A] time = 0.102331, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{5 x+3}+\frac{1331}{512} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]*(3 + 5*x)^(5/2),x]
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Rubi in Sympy [A] time = 9.76436, size = 104, normalized size = 0.9 \[ \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{20} - \frac{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{240} - \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{384} - \frac{1331 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{512} + \frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{5120} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(5/2)*(1-2*x)**(1/2),x)
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Mathematica [A] time = 0.0728457, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9600 x^3+15520 x^2+5836 x-4005\right )-43923 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15360} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]*(3 + 5*x)^(5/2),x]
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Maple [A] time = 0.008, size = 104, normalized size = 0.9 \[{\frac{1}{20} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{11}{240} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{121}{384} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{512}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{14641\,\sqrt{10}}{10240}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(5/2)*(1-2*x)^(1/2),x)
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Maxima [A] time = 1.48436, size = 95, normalized size = 0.82 \[ -\frac{5}{8} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{91}{96} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{605}{128} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{14641}{10240} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{121}{512} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.217327, size = 90, normalized size = 0.78 \[ \frac{1}{30720} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (9600 \, x^{3} + 15520 \, x^{2} + 5836 \, x - 4005\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 43923 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1),x, algorithm="fricas")
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Sympy [A] time = 37.5278, size = 272, normalized size = 2.34 \[ \begin{cases} \frac{125 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{10 x - 5}} - \frac{1925 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{24 \sqrt{10 x - 5}} - \frac{605 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{192 \sqrt{10 x - 5}} - \frac{6655 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{768 \sqrt{10 x - 5}} + \frac{14641 i \sqrt{x + \frac{3}{5}}}{512 \sqrt{10 x - 5}} - \frac{14641 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5120} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5120} - \frac{125 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{- 10 x + 5}} + \frac{1925 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{24 \sqrt{- 10 x + 5}} + \frac{605 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{192 \sqrt{- 10 x + 5}} + \frac{6655 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{768 \sqrt{- 10 x + 5}} - \frac{14641 \sqrt{x + \frac{3}{5}}}{512 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)**(5/2)*(1-2*x)**(1/2),x)
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GIAC/XCAS [A] time = 0.245229, size = 220, normalized size = 1.9 \[ \frac{1}{76800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1),x, algorithm="giac")
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