Optimal. Leaf size=142 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{15 (5 x+3)^{3/2}}-\frac{524 \sqrt{1-2 x} (3 x+2)^3}{825 \sqrt{5 x+3}}+\frac{623 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (8940 x+2563)}{220000}+\frac{35511 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20000 \sqrt{10}} \]
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Rubi [A] time = 0.26437, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{15 (5 x+3)^{3/2}}-\frac{524 \sqrt{1-2 x} (3 x+2)^3}{825 \sqrt{5 x+3}}+\frac{623 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (8940 x+2563)}{220000}+\frac{35511 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(2 + 3*x)^4)/(3 + 5*x)^(5/2),x]
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Rubi in Sympy [A] time = 26.781, size = 133, normalized size = 0.94 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{524 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{825 \sqrt{5 x + 3}} + \frac{623 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{1375} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{704025 x}{2} + \frac{807345}{8}\right )}{1237500} + \frac{35511 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{200000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**4*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
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Mathematica [A] time = 0.210641, size = 70, normalized size = 0.49 \[ \frac{\frac{10 \sqrt{1-2 x} \left (3564000 x^4+8999100 x^3+6384015 x^2+995870 x-218953\right )}{(5 x+3)^{3/2}}-1171863 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6600000} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^4)/(3 + 5*x)^(5/2),x]
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Maple [A] time = 0.021, size = 147, normalized size = 1. \[{\frac{1}{13200000} \left ( 71280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+29296575\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+179982000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+35155890\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+127680300\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+10546767\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +19917400\,x\sqrt{-10\,{x}^{2}-x+3}-4379060\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.222463, size = 127, normalized size = 0.89 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (3564000 \, x^{4} + 8999100 \, x^{3} + 6384015 \, x^{2} + 995870 \, x - 218953\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1171863 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{13200000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**4*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.315704, size = 255, normalized size = 1.8 \[ \frac{27}{500000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 5 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 475 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{8250000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{35511}{200000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{263 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{687500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{789 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{515625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="giac")
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